Information, Liquidity, and
Dynamic Limit Order Markets∗
Roberto Ricco† Barbara Rindi‡ Duane J. Seppi§
March 11, 2018
Abstract
This paper describes price discovery and liquidity provision in a dynamic limit order market
with asymmetric information and non-Markovian learning. In particular, investors condition
on information in both the current limit order book and on the prior trading history when
deciding whether to provide or take liquidity. In addition, we show that the information content
of arriving orders can depend crucially on the size of value shocks relative to the discreteness
in the price grid. When information shocks are small, the information content of orders can be
non-monotone both in the direction and aggressiveness of arriving orders.
JEL classification: G10, G20, G24, D40
Keywords: Limit order markets, asymmetric information, liquidity, market microstructure
∗We thank Sandra Fortini, Thierry Foucault, Paolo Giacomazzi, Burton Hollifield, Phillip Illeditsch, Stefan Lewel-len, Marco Ottaviani, Tom Ruchti, and seminar participants at Carnegie Mellon University for helpful comments.We are grateful to Fabio Sist for his significant contribution to the computer code for our model.
†Bocconi University. Phone: +39-02-5836-2715. E-mail: [email protected]‡Bocconi University and IGIER. Phone: +39-02-5836-5328. E-mail: [email protected]§Tepper School of Business, Carnegie Mellon University. Phone: 412-268-2298. E-mail: [email protected]
The aggregation of private information and the dynamics of liquidity supply and demand are closely
intertwined in financial markets. In dealer markets, informed and uninformed investors trade via
market orders and, thus, take liquidity, while dealers provide liquidity and try to extract information
from the arriving order flow (as in Kyle (1985) and Glosten and Milgrom (1985)). However, in limit
order markets — the dominant form of securities market organization today — the relation between
who has information and who is trying to learn it and who supplies and demands liquidity is not
well understood theoretically.1 Recent empirical research highlights the role of informed traders
not only as liquidity takers but also as liquidity suppliers. O’Hara (2015) argues that fast informed
traders use market and limit orders interchangeably and often prefer limit orders to marketable
orders. Fleming, Mizrach, and Nguyen (2017) and Brogaard, Hendershott, and Riordan (2016) find
that limit orders play a significant empirical role in price discovery.2
Our paper presents the first rational expectations model of a dynamic limit order market with
asymmetric information and history-dependent Bayesian learning. In particular, learning is not
constrained to be Markovian. The model represents a trading day with market opening and closing
effects. Our model lets us investigate the information content of different types of market and limit
orders, the dynamics of who provides and demands liquidity, and the non-Markovian information
content of the trading history. In addition, we study how changes in the amount of adverse selection
— in terms of both the asset-value volatility and the arrival probability of informed traders — affect
liquidity, price discovery, and welfare. We have three main results:
• Increased adverse selection does not always worsen market liquidity as in Kyle (1985). Liquid-
ity can potentially improve if informed traders with better information trade more aggressively
by submitting limit-orders at the inside quotes rather than using market orders.
1See Jain (2005) for a discussion of the prevalence of limit order markets. See Parlour and Seppi (2008) for asurvey of theoretical models of limit order markets. See Rindi (2008) for a model of informed traders as liquidityproviders.
2Gencay, Mahmoodzadeh, Rojcek, and Tseng (2016) investigate brief episodes of high-intensity/extreme behaviorof quotation process in the U.S. equity market (bursts in liquidity provision that happen several hundreds of timea day for actively traded stocks) and find that liquidity suppliers during these bursts significantly impact prices byposting limit orders.
1
• The relation between limit and market orders and their information content depends on the
size of private information shocks relative to the tick size. Indeed, the information content of
orders can even be opposite the order direction and aggressiveness.
• The learning dynamics are non-Markovian in that the trading history has information in
addition to the current state of the limit order book. In addition, the incremental information
content of arriving limit and market orders is history-dependent.
Dynamic limit order markets with uninformed investors are studied in a large literature. This
includes Foucault (1999), Parlour (1998), Foucault, Kadan, and Kandel (2005), and Goettler,
Parlour, and Rajan (2005). There is some previous theoretical research that allows informed traders
to supply liquidity. Kumar and Seppi (1994) is a static model in which optimizing informed and
uninformed investors use profiles of multiple limit and market orders to trade. Kaniel and Liu
(2006) extend the Glosten and Milgrom (1985) dealership market to allow informed traders to post
limit orders. Aıt-Sahalia and Saglam (2013) also allow informed traders to post limit orders, but
they do not allow them to choose between limit and market orders. Moreover, the limit orders
posted by their informed traders are always at the best bid and ask prices. Goettler, Parlour,
and Rajan (2009) allow informed and uninformed traders to post limit or market orders, but their
model is stationary and assumes Markovian learning. Rosu (2016b) studies a steady-state limit
order market equilibrium in continuous-time with Markovian learning and additional information-
processing restrictions. These last two papers are closest to ours. Our model differs from them
in two ways: First, they assume Markovian learning in order to study dynamic trading strategies
with order cancellation, whereas we simplify the strategy space (by not allowing dynamic order
cancellations and submissions) in order to investigate non-Markovian learning (i.e., our model has
a larger state space). Second, we model a non-stationary trading day with opening and closing
effects and history-dependent Bayesian learning. Market opens and closes are important daily
features of stock markets. Bloomfield, O’Hara, and Saar (2005) show in an experimental asset
market that informed traders sometimes provide more liquidity than uninformed traders. Our
model provides equilibrium examples of liquidity provision by informed investors.
A growing literature investigates the relation between information and trading speed (e.g., Biais,
2
Foucault, and Moinas (2015); Foucault, Hombert, and Rosu (2016); and Rosu (2016a)). However,
these models assume Kyle or Glosten-Milgrom market structures and, thus, cannot consider the
roles of informed and uninformed traders as endogenous liquidity providers and demanders. We
argue that understanding price discovery dynamics in limit order markets is an essential precursor
to understanding speedbumps and cross-market competition given the real-world prevalence of limit
order markets.
1 Model
We consider a limit order market in which a risky asset is traded at five times tj ∈ {t1, t2, t3, t4, t5}
over a trading day. The fundamental value of the asset after time t5 at the end of the day is
v = v0 + ∆ =
v = v0 + δ with Pr(v) = 1
3
v0 with Pr(v0) = 13
v¯
= v0 − δ with Pr(v¯) = 1
3
(1)
where v0 is the ex ante expected asset value, and ∆ is a symmetrically distributed value shock. The
limit order market allows for trading through two types of orders: Limit orders are price-contingent
orders that are collected in a limit order book. Market orders are executed immediately as the best
available price in the limit order book. The limit order book has a price grid with four prices,
Pi = {A2, A1, B1, B2}, two each on the ask and bid sides of the market. The tick size is equal to
κ > 0, and the ask prices are A2 = v0 + κ, A1 = v0 + κ2 ; and by symmetry the bid prices are
B2 = v0 − κ, B1 = v0 − κ2 . Order execution in the limit order book follows time and price priority.
Investors arrive sequentially over time to trade in the market. At each time tj one investor
arrives. Investors are risk-neutral and asymmetrically informed. A trader is informed with prob-
ability α and uninformed with probability 1− α. Informed investors know the realized asset-value
shock ∆ perfectly. Uninformed investors do not know ∆, but they use Bayes’ Rule and their know-
ledge of the equilibrium to learn about ∆ from the observable market dynamics over time. An
investor arriving at time tj may also have a personal private-value trading motive, which — we
3
assume for tractability — causes them to adjust their valuation of v0 to βtjv0 where the factor βtj
may be greater than or less than 1. Non-informational private-value motives include preference
shocks, hedging needs, and taxation. The absence of a non-informational trading motive would
lead to the Milgrom and Stokey (1982) no-trade result. The factor βtj at time tj is drawn from
a truncated normal distribution, Tr[N (µ, σ2)], with support over the interval [0, 2]. The mean is
µ = 1, which corresponds to a neutral private valuation. Traders with neutral valuations tend to
provide liquidity symmetrically on both the buy and sell sides of the market, while traders with
extreme private valuations provide one-sided liquidity or actively take liquidity. The parameter σ
determines the dispersion of a trader’s private-value factor βtj , as shown in Figure 1, and, thus,
the probability of large private gains-from-trade due to extreme investor private valuations.
The sequence of arriving investors is independently and identically distributed in terms of
whether they are informed or uninformed and in terms of their individual gains-from-trade factors
βtj . In one specification of our model, only uninformed investors have private valuations, while in
a second richer specification both informed and uninformed investors have private valuations. A
generic informed investor is denoted as I, where we denote the informed investor as Iv if the value
shock is positive (∆ = δ), as Iv¯
if the shock is negative (∆ = −δ), and as Iv0 if the shock is is
zero (∆ = 0). Informed investors arriving at different times during the day all have the identical
asset-value information. Uninformed investors are denoted as U .
An investor arriving at time tj can take one of seven possible actions xtj : One possibility is
to submit a buy or sell market order MOAi,tj or MOBi,tj to buy or sell immediately at the best
available ask or bid respectively in the limit order book at time tj . A subscript i = 1 indicates that
the best quote at time tj is at the inside quote, and i = 2 means the best quote is at the outside
quote. Alternatively, the investor can submit one of four possible limit orders LOAi,tj and LOBi,tj
on the ask or bid side of the book, respectively. A subscript i = 1 denotes an aggressive limit order
posted at the inside quote, and i = 2 is a less aggressive limit order at the outside quote. Yet
another alternative is to choose to do nothing (NTtj ).
For tractability, we make a few simplifying assumptions. Limit orders cannot be modified or
canceled after submission. Thus, each arriving investor has one and only one opportunity to submit
4
Figure 1: Distribution of Traders’ Private-Value Factors - β ∼ Tr[N (µ, σ2)].This figure shows the truncated Normal probability density Function (PDF) of trader private-value factors βtj witha mean µ = 1 and three different values of dispersion σ.
0.0 0.5 1.0 1.5 2.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Private-Value Factor
Density
σ=1
σ=1.5
σ=2
an order. There is also no quantity decision. Orders are to buy or sell one share. Lastly, investors
can only submit one order. Taken together, these assumptions let us express the traders’ action
space as Xtj = {MOBi,tj , LOA1,tj , LOA2,tj , NTtj , LOB2,tj , LOB1,tj ,MOAi,tj}, where each of the
orders denotes an order for one share.
In addition to the arriving informed and uninformed traders, there is a market-making trading
crowd that submits limit orders to provide liquidity. By assumption, the crowd just posts single
limit orders at the outside prices A2 and B2. The market opens with an initial book submitted
by the crowd at time t0. After each subsequent order-submission time tj for arriving informed and
uninformed traders, the crowd replenishes the book at the outside prices, if needed, when either
side of the book is empty. If there are still limit orders at prices A2 and B2 on both sides of the
book, then the crowd does not submit any limit orders. For tractability, we assume that public
limit orders by the arriving informed and uninformed investors have priority over limit orders from
the crowd. The focus of our model is on market dynamics involving information and liquidity given
the behavior of optimizing informed and uninformed investors. The crowd is simply a modeling
device to insure it is always possible for arriving traders to submit market orders if they so choose.
5
Market dynamics over the trading day are intentionally non-stationary in our model in order
to capture market opening and closing effects. When the market opens at t1 there are no standing
limit orders in the book except from those at prices A2 or at B2 from the trading crowd.3 At the
end of the day all unexecuted limit orders are cancelled.
The state of the limit order book at time tj given orders from arriving investors is
Ltj = [qA2tj, qA1tj, qB1tj, qB2tj
] (2)
where qAitj
and qBitj
indicate the depth at prices Ai and Bi at time tj . In addition, there are limit
orders from the crowd. While the crowd’s orders are in the book, we net them out when talking
about the informational “state” of the book, since they are perfectly predictable. Let ∆Ltj be the
change in the limit order book generated by an arriving informed and uninformed investor’s action
xtj ∈ Xtj at time tj :4
∆Ltj = [∆qA2tj,∆qA1
tj,∆qB1
tj,∆qB2
tj] =
[−1, 0, 0, 0] if xtj = MOA2,tj
[0,−1, 0, 0] if xtj = MOA1,tj
[+1, 0, 0, 0] if xtj = LOA2,tj
[0,+1, 0, 0] if xtj = LOA1,tj
[0, 0, 0, 0] if xtj = NT
[0, 0,+1, 0] if xtj = LOB1,,tj
[0, 0, 0,+1] if xtj = LOB2,tj
[0, 0,−1, 0] if xtj = MOB1,tj
[0, 0, 0,−1] if xtj = MOB2,tj
(3)
where “+1” with a limit order denotes the addition of an order at a particular limit price and “−1”
denotes execution of an earlier BBO limit order in the book. The resulting dynamics of the limit
3In practice, daily opening limit order books include uncancelled orders from the previous day and new limitorders from opening auctions. For simplicity, we abstract from these interesting features of markets.
4There are nine alternatives in (3) because we allow separately for cases in which the best bid and ask for marketsells and buys are at the inside and outside quotes.
6
order book are:
Ltj = Ltj−1 + ∆Ltj (4)
where j = 1, . . . , 5. An important source of information in our model is the observed trading
history of orders posted at times t1, .., tj in the market. We denote an order-flow history by
Ltj = {∆Lt1 , . . . ,∆Ltj}. When traders arrive in the market, they observe the history of market
activity up through the current standing limit order book at the time they arrive.
Investors trade using optimal order-submission strategies given their information and any private-
value motive. If an uninformed investor arrives at time tj , then his order xtj is chosen to maximize
his expected terminal payoff
maxx∈Xtj
ϕU (x |βtj ,Ltj−1) = E[(βtj v0 + ∆− p(x)) f(x)|βtj ,Ltj−1 ] (5)
= [βtj v0 + E[∆ |Ltj−1 , θxtj ]− p(x)]Pr(θxtj |Ltj−1)
where p(x) is the price at which order x trades, and f(x) denotes the amount of the submitted order
that is actually “filled.” If x is a market order, then f(x) = 1 (i.e., all of the order is executed),
and the execution price p(x) is the best quote on the other side of the book at time tj . If x is a
non-marketable limit order, then the execution price p(x) is its limit price, but the fill amount f(x)
is random variable equal to 1 if the limit order is filled and zero if it is not filled. If the investor
does not trade — either because no order is submitted or because a limit order is not filled —
then f(x) is zero. In the second line of (5), the expression θxtj denotes the set of future trading
states at times tj+1, . . . , t5 in which the order x is executed. This matters because the sequence
of subsequent orders in the market, which may or may not result in the execution of a limit order
submitted at time tj , is correlated with the asset value shock ∆. For example, future market buy
orders are more likely if the ∆ shock is positive (since Iv investors will want to buy). Uninformed
investors rationally take the relation between future orders and ∆ into account when forming their
expectation E[∆ |Ltj−1 , θxtj ] of what the asset will be worth in states in which their limit orders
are executed. The second line of (5) also makes clear that uninformed investors use the prior
order history Ltj−1 in two ways: It affects their beliefs about limit order execution probabilities
7
Pr(θxtj |Ltj−1) and their execution-state-contingent asset-value expectations E[∆ |Ltj−1 , θxtj ].
An informed investor who arrives at tj chooses an order xtj to maximize her expected payoff
maxx∈Xtj
ϕI(x | v, βtj ,Ltj−1) = E[(βtj v0 + ∆− p(x)) f(x)| v, βtj ,Ltj−1 ] (6)
= [βtj v0 + ∆− p(x)]Pr(θxtj | v,Ltj−1)
The only uncertainty for informed investors is about whether any limit orders they submit will be
executed. Their belief about this probability Pr(θxtj | v,Ltj−1) is conditioned on both the trading
history up through the current book and on their knowledge about the ending asset value. Thus,
informed traders condition on Ltj−1 , not to learn about ∆ (which they already know) or about
future private-value factors βtj (which are i.i.d. over time), but because they understand that
the trading history is an input in the trading behavior of future uninformed investors with whom
they might trade in the future. Our analysis considers two model specifications for the informed
investors. In one, informed investors have no private-value motive, so that their β factors are equal
to 1. In the second specification, their β factors are random and are independently drawn from the
same truncated normal distribution Tr[N (µ, σ2)] as the uninformed investors.
The optimization problem in (5) defines sets of actions xtj ∈ Xtj that are optimal for the
uninformed investor at different times tj given different private-value factors β and order histories
Ltj−1 . These optimal orders can be unique or there may be multiple orders which make the
uninformed investor equally well-off. The optimal order-submission strategy for the uninformed
investor is a probability function γUj (x|β,Ltj−1) that is zero if the order x is suboptimal and equals
a mixing probability over optimal orders. If an optimal order x is unique, then γj(x|β,Ltj−1) = 1.
Similarly, the optimization problem in (6) can be used to define an optimal order submission
strategy γIj (x|β, v,Ltj−1) for informed investors at time tj given their factor β, their knowledge
about the asset value v, and the order history Ltj−1 .
8
1.1 Equilibrium
An equilibrium is a set of mutually consistent optimal strategy functions and beliefs for uninformed
and informed investors for each time tj , given each order history Ltj−1 , private-value factor βtj ,
and (for informed traders) private information v. This section explains what “mutually consistent”
means and then gives a formal definition of an equilibrium in our model.
A central feature of our model is asymmetric information. The presence of informed traders
means that, by observing prices and associated quantities (i.e., past and current states of the book),
uninformed traders can infer information about the asset value and use it in their order-submission
strategies. More precisely, uninformed traders rationally learn from the trading history about the
probability that the future value of the asset will go up, stay constant, or go down. However,
investors cannot learn about the private values (β) or information status (I or U) of future traders
since these are both i.i.d over time. Informed traders do not need to learn about v since they know
it. However, they do condition their trading behavior on v (since that tells them what types of
informed traders will arrive in the future along with the uninformed traders) and they condition
on the trading history (since that is informative about the trading behavior of future uninformed
traders since the trading history is an input in their order-submission strategy functions).
The underlying economic state in our model is the realization of the asset value v and a realized
sequence of investors who arrive in the market. The investor who arrives at time tj is described by
two characteristics: their status as being informed or uninformed, Iv or U , and their private-value
factor β. The underlying economic state is exogenously chosen over time by Nature. More formally,
it follows an exogenous stochastic process described by the model parameters δ, α, µ, and σ. A
sequence of arriving investors together with a pair of strategy functions — which we denote here
as Γ = {γUj (x|β,Ltj−1), γIj (x|β, v,Ltj−1)} — induce a sequence of trading actions which results in
a sequence of observable changes in the state of the limit order book. Thus, the stochastic process
generating paths of trading outcomes (i.e., trading histories in the limit order book) is induced by
the economic state process and the strategy functions. Given the trading-outcome path process,
there are several things we can compute directly: First, we can compute the unconditional prob-
abilities of different paths Pr(Ltj ) and the conditional probabilities Pr(∆Ltj |Ltj−1) of particular
9
order book changes ∆Ltj given a prior history Ltj−1 . In particular, we can identify paths of trad-
ing outcomes that are possible (i.e., have positive probability Pr(Ltj )) given the strategy functions
{γUj (x|β,Ltj−1), γIj (x|β, v,Ltj−1)} and paths of trading outcomes which are not possible (i.e., for
which Pr(Ltj ) = 0). Second, the trading-outcome path process also determines the probabilities
Pr(θxtj | v,Ltj−1) and Pr(θxtj |Ltj−1) for informed and uninformed investors — at any given time tj
given a prior trading history and an investor’s information — that a limit order x submitted at
time tj will be executed in the future. Computing each of these probabilities is simply a matter of
listing all of the possible underlying economic states, mechanically applying the order-submission
rules, identifying the relevant outcomes path-by-path, and then taking expectations across paths.
Let ` denote the set of all physically feasible order histories {Ltj : j = 1, . . . , 4} of lengths up to
four trading periods. A four-period long history is the longest history a order-submission strategy
can depend on in our model. In this context, feasible paths are simply sequences of actions in
the action choice set without regard to whether they are possible in the sense that they can occur
with positive probability given the strategy functions Γ. Let ` in,Γ denote the subset of all possible
trading paths in ` that have positive probability, Pr(Ltj ) > 0 given a pair of order strategies Γ. Let
` off,Γ denote the complementary set of trading paths that are feasible but not possible given Γ. This
notation will be useful when discussing “off equilibrium” beliefs. In our analysis, strategy functions
Γ are defined for all feasible paths in `. In particular, this includes all of the possible paths in ` in,Γ
given Γ and also the paths in ` off,Γ. As a result, the probabilities Pr(∆Ltj |Ltj−1), Pr(θxtj | v,Ltj−1)
and Pr(θxtj |Ltj−1) are always well-defined, because the continuation trading process going forward,
even after an unexpected order-arrival event (i.e., a path Ltj−1 ∈ ` off,Γ), is still well-defined.
The stochastic process for trading-outcome paths and its relation to the underlying economic
state also determine the uninformed-investor expectations E[v |Ltj , θxtj ] of the terminal asset value
given the previous order history (Ltj ) and conditional on future limit-order execution (θxtj ). These
expectations are determined as follows:
• Step 1: The conditional probabilities πvtj = Pr(v|Ltj ) of a particular final asset value v = v, v0
or v given a possible trading history Ltj ∈ ` in,Γ up through time tj is given by Bayes’ Rule.
10
At time t1, this probability is
πvt1 =Pr(v,Lt1)
Pr(Lt1)=Pr(Lt1 |v)Pr(v)
Pr(Lt1)=Pr(∆Lt1 |v)Pr(v)
Pr(∆Lt1)(7)
=Pr(∆Lt1 |v, I)Pr(I) + Pr(∆Lt1 |U)Pr(U)
Pr(∆Lt1)Pr(v)
=Eβ[γI1(xt1 |βIt1 , v)|v]α+ Eβ[γU1 (xt1 |βUt1)](1− α)
Pr(∆Lt1)πvt0
where the prior is the unconditional probability πvt0 = Pr(v), xt1 is the trading action at time
tj that leads to the order book change ∆Lt1 , and βIt1 and βUt1 are independently distributed
private-value β realizations for informed and uninformed investors at time t1.5 At time tj > t1,
this probability is given recursively by6
πvtj =Pr(v,Ltj )
Pr(Ltj )=Pr(v,∆Ltj ,Ltj−1)
Pr(∆Ltj ,Ltj−1)(8)
=
Pr(∆Ltj |v,Ltj−1 , I)Pr(I|Ltj−1)Pr(v|Ltj−1)
+Pr(∆Ltj |v,Ltj−1 , U)Pr(U |Ltj−1)Pr(v|Ltj−1)
Pr(∆Ltj |Ltj−1)
=Eβ[γIj (xtj |βItj , v,Ltj−1)|v,Ltj−1 ] α+ Eβ[γUj (xtj |βUtj ,Ltj−1)|Ltj−1 ] (1− α)
Pr(∆Ltj |Ltj−1)πvtj−1
These probabilities can then be used to compute the uninformed investor expected asset value
conditional on the order history path
E[v|Ltj−1 ] = πvtj−1v + πv0
tj−1v0 + π
vtj−1
v (9)
• Step 2: The conditional probabilities πvtj given a “feasible but not possible in equilibrium”
order history Ltj ∈ ` off,Γ in which a limit order book change ∆Ltj that is inconsistent with
5A trader’s information status (I or U) is independent of the asset value v, so P (I|v) = Pr(I) and Pr(U |v) =Pr(U). Furthermore, uninformed traders have no private information about v, so the probability Pr(∆Lt1 |U) withwhich they take a trading action ∆Lt1 does not depend on v.
6A trader’s information status is again independent of v, and it is also independent of the past trading historyLt1 . While the probability with which an uninformed trader takes a trading action ∆Lt1 may depend on the pastorder history Ltj , it does not depend directly on v which uninformed traders do not know.
11
the strategies Γ at time tj are set as follows:
1. If the priors are fully revealing in that πvtj−1= 1 for some v, then πvtj = πvtj−1
for all v.
2. If the priors are not fully revealing at time tj , then πvtj = 0 for any v for which πvtj−1= 0
and the probabilities πvtj for the remaining v’s can be any non-negative numbers such
that πvtj + πv0tj
+ πvtj
= 1.
3. Thereafter, until any next unexpected trading event, the subsequent probabilities πvtj′
for j′ > j are updated according to (8).
• Step 3: The execution-contingent conditional probabilities πvtj = Pr(v|Ltj−1 , θxtj ) of a final
asset value v conditional on a prior path Ltj−1 and on execution of a limit order x submitted
at time tj is
πvtj =Pr(Ltj−1)Pr(v|Ltj−1) Pr(θxtj−1
|v,Ltj−1)
Pr(θxtj ,Ltj−1)(10)
=Pr(θxtj |v,Ltj−1)
Pr(θxtj |Ltj−1)πvtj−1
This is true when adjusting for a future execution contingency when the probabilities πvtj−1
given the prior history Ltj−1 are for possible paths in ` in,Γ (from (7) and (8) in Step 1) and
also for feasible but not possible paths in ` off,Γ (from Step 2). These execution-contingent
probabilities πvtj are used to compute the execution-contingent conditional expected asset
value
E[v|Ltj−1 , θxtj ] = πvtj v + πv0
tjv0 + π
vtjv¯
(11)
used by uninformed traders to compute the expected payoffs for limit orders. In particular,
note that these are the execution-contingent probabilities πvtj from (10) rather than the prob-
abilities πvtj from (8) that just condition on the prior trading history but not on the future
states in which the limit order is executed.
Given these updating dynamics, we can now define an equilibrium.
Definition. A Perfect Bayesian Nash Equilibrium of the trading game in our model is a col-
12
lection {γU, ∗j (x|β,Ltj−1), γI, ∗j (x|β, v,Ltj−1), P r∗(θxtj | v,Ltj−1), P r∗(θxtj |Ltj−1), E∗[v|Ltj−1 , θxtj ]} of
order-submission strategies, execution-probability functions, and execution-contingent conditional
expected asset-value functions such that:
• The equilibrium execution probabilities Pr∗(θxtj | v,Ltj−1) and Pr∗(θxtj |Ltj−1) are consist-
ent with the equilibrium order-submission strategies {γU, ∗j+1(x|β,Ltj ), . . . , γU, ∗5 (x|β,Lt4)} and
{γI, ∗j+1(x|β, v,Ltj ), . . . , γI, ∗5 (x|β, v,Lt4)} after time tj .
• The execution-contingent conditional expected asset values E∗[v|Ltj−1 , θxtj ]} agree with Bayesian
updating equations (7), (8), (10), and (11) in Steps 1 and 3 when the order x is consistent with
the equilibrium strategies γU, ∗j (x|β,Ltj−1) and γI, ∗j (x|β, v,Ltj−1) at date tj and, when x is
an off-equilibrium action inconsistent with the equilibrium strategies, with the off-equilibrium
updating in Step 2.
• The positive-probability supports of the equilibrium strategy functions γU, ∗j (x|β,Ltj−1) and
γI, ∗j (x|β, v,Ltj−1) (i.e., the orders with positive probability in equilibrium) are subsets of
the sets of optimal orders for uninformed and informed investors computed from their op-
timization problems (5) and (6) and the equilibrium execution probabilities and outcome-
contingent conditional asset-value expectation functions Pr∗(θxtj | v,Ltj−1), Pr∗(θxtj |Ltj−1),
and E∗[v|Ltj−1 , θxtj ].
The Appendix explains the algorithm used to compute the equilibria in our model. To help with
intuition, the next section walks through the order-submission and Bayesian updating mechanics
for a particular path in the extensive form of the model.
Our equilibrium concept differs from the Markov Perfect Bayesian Equilibrium used in Goettler
et al. (2009). Beliefs and strategies in our model are path-dependent; traders use Bayes Rule
given the full prior order history when they arrive in the market. In contrast, Goettler et al.
(2009) restricts Bayesian updating to the current state of the limit order book but do not allow for
conditioning on the previous order history. Rosu (2016b) also assumes a Markov Perfect Bayesian
Equilibrium. The quantitative importance of the order history is an issue that is considered when
we discuss our results in Section 2.
13
1.2 Illustration of order-submission mechanics and Bayesian updating
This section uses an excerpt of the extensive form of the trading game in our model to illustrate
order-submission and trading dynamics and the associated Bayesian updating process. The partic-
ular trading history path in Figure 2 is from the equilibrium for the model specification in which
both informed and uninformed investors have private-value motives. The parameter values are
σ = 1.5, α = 0.8, and δ = 0.16, which is a market with a relatively high informed investor ar-
rival probability and large information shocks. In this illustration, Nature has chosen an economic
state in which there will be good news (v) about the asset, and the arriving sequence of traders
considered here is {I, U, U, I, I}. Trading starts at t1 with an empty public book, Lt0 = [0, 0, 0, 0],
and the limit orders from the trading crowd at prices A2 and B2. For simplicity, our discussion
here only reports a few nodes of the trading game with their associated equilibrium strategies. For
example, we do not include NT at the end of t1, since, as we show later in the paper, NT is not
an equilibrium action at t1 for these parameters.
The path in Figure 2 can also be used to illustrate the Bayesian updating dynamics in the
model. After the investor at t1 has been observed submitting a limit order LOA2,t1 at time t1, the
uninformed trader who arrives in this example at time t2 — who just knows the submitted order
at time t1 but not the identity or information of the trader at time t1 — updates his equilibrium
conditional valuation to be E[v|LOA2,t1 ] = 1.056 and his execution-contingent expectation given
his limit order LOA1,t2 at time t2 to be E[v|LOA2,t1 , θLOA1,t2 ] = 1.089.7 In subsequent periods,
traders observe additional realized orders and then further update their beliefs.
Traders in our equilibrium choose from a discrete number of possible orders given their respective
information and any private value trading motives. In the equilibrium path considered here, the
optimal strategies do not involve any randomization across different orders. Optimal orders are
unique given the inputs. Figure 2 shows below each order type at each time the probabilities with
which the different orders are submitted by the trader who arrived. For example, if an informed
trader Iv arrives at t1, she chooses a limit order LOA2,t1 to sell at A2 with probability 0.118.8 Each
7The numerical values of these expectations are taken from our equilibrium calculations.8The value of this order submission probability and others mentioned in the rest of this section are taken from
the computation of the equilibrium.
14
of these unique optimal order is associated with a different range of β types (for both informed and
uninformed investors) and value signals (for informed investors). Figure 3 illustrates where these
order submission probabilities come from by superimposing the upper envelope of the expected
payoffs for the different optimal orders at time t1 on the β distribution. It shows how different β
ranges correspond to a discrete set of equilibrium strategies delimited by the β thresholds. At each
trading time, as the trading game progresses along this path, traders submit orders (or do not trade)
following their equilibrium order-submission strategies. The equilibrium execution probabilities of
their orders depend on the order-submission decisions of future traders, which, in turn, depend on
their trading strategies and the input information (i.e., their β realizations, any private knowledge
about v, and the trading history path at the times they arrive). At time t1, the initial trader has
rational-expectation beliefs that the execution probability of her LOA2,t1 order posted at t1 is 0.644
(see Table 3). This equilibrium execution probability depends on all of the possible future trading
paths from the submission time t1 up through time t5. For example, one possibility is that the
LOA2,t1 order will be hit by an investor arriving at time t2 who submits a market order. Another
possibility (which is what happens along this particular path) is that the next period (at t2) an
uninformed trader could arrive and post a limit order LOA1,t2 to sell at A1, thereby undercutting
the LOA2,t1 order — so that the book is Lt2 = [1, 1, 0, 0]) at the end of t2. In this scenario, the
initial LOA2,t1 order from t1 will only be executed provided that the LOA1,t2 order submitted at
t2 is executed first. For example, the probability of a market order MOA1,t3 hitting the limit order
at A1 at t3 is 0.365, and then the probability of another market order hitting the initial limit sell
at A2 is 0.423 at t4 or 0.505 at t5.9 Therefore, there is a chance that the LOA2,t1 order from t1 will
still be executed if it is undercut by an order LOA1,t2 at t2.
9Due to space constraint we cannot include the t4 node in Figure 2.
15
2 Results
Our analysis investigates how liquidity supply and demand decisions of informed and uninformed
traders and the learning process of uninformed traders affect market liquidity, price discovery, and
investor welfare. This section presents numerical results for our model. We first consider a model
specification in which only uninformed investors have a random private-value trading motive. In a
second specification, we generalize the analysis and show the robustness of our findings and extend
them. The tick size κ is fixed at 0.10, and the private-value dispersion σ is 1.5 throughout. We
investigate comparative statics for the amount of adverse selection. We also show that our model
has significant non-Markovian learning that would be missed in constrained Markovian equilibria.
Our analysis focuses on two time windows. The first is when the market opens at time t1. The
second is over the middle of the trading day from times t2 through t4. We look at these two windows
because our model is non-stationary over the trading day. Much like actual trading days, our model
has start-up effects at the beginning of the day and terminal horizon effects at the market close.
When the market opens at time t1, there are time-dependent incentives to provide rather than to
take liquidity: The incoming book is thin (with limit orders only from the crowd), and there is
the maximum amount of time for future investors to arrive to hit limit orders from t1. There are
also time-dependent disincentives to post limit orders. Information asymmetries are maximal at
t1, since there has been no learning from the trading process. Over the day, information is revealed
(lessening adverse selection costs), but the book can become fuller (i.e., there is competition in
liquidity provision from earlier limit orders), and the remaining time for market orders to arrive
and execute limit orders becomes shorter. Comparing these two time windows shows how market
dynamics change over the day. The market close at t5 is also important, but trading then is
straightforward. At the end of the day, investors only submit market orders, because the execution
probability for new limit orders submitted at t5 is zero given our assumption that unfilled limit
orders are canceled once the market closes.
We use our model to investigate three questions: First, who provides and takes liquidity, and how
does the amount of adverse selection affect investor decisions to take and provide liquidity? Second,
how does market liquidity vary with different amounts of adverse selection? Third, how does the
16
Figure 2: Excerpt of the Extensive Form of the Trading Game. This figure shows one of thepossible trading paths of the trading game with parameters α = 0.8, δ = 0.16, µ = 1, σ = 1.5, κ = 0.10, and 5 timeperiods. Before trading starts (t0) with an empty book (0, 0, 0, 0) at all price levels (Ai and Bi with i = {1, 2}), Natureselects v = {v, v0, v} with probabilities { 1
3, 1
3, 1
3}. At each trading period nature also selects an informed trader (I)
with probability α and an uninformed trader (U) with probability (1−α). Arriving traders choose the optimal orderat each period which may potentially include limit orders LOAt (LOBt) or market orders at the best ask, MOAi,t,or at the best bid, MOBi,t. Below each optimal trading strategy we report in italics its equilibrium order-submissionprobability. Boldfaced equilibrium strategies and associated states of the book (within double vertical bar) indicatethe states of the book that we consider at each node of the chosen trading path.
tj = t0A2 0 TCA1 0B1 0B2 0 TC
v
tj = t1I
MOA2 LOB1 LOB2 LOA2 LOA1 MOB2
0.256 0.282 0.030 0.118 0.314 0.0000 0 0 ‖ 1 ‖ 0 00 0 0 ‖ 0 ‖ 1 00 1 0 ‖ 0 ‖ 0 00 0 1 ‖ 0 ‖ 0 0
tj = t2
I...
α
tj = t2U
MOA2 LOB1 LOB2 LOA2 LOA1 MOB2
0.164 0.296 0.083 0.000 0.457 0.0000 1 1 2 ‖ 1 ‖ 10 0 0 0 ‖ 1 ‖ 00 1 0 0 ‖ 0 ‖ 00 0 1 0 ‖ 0 ‖ 0
tj = t3I...
α
tj = t3U
MOA1 LOB1 LOB2 LOA2 LOA1 MOB2
0.365 0.000 0.219 0.000 0.058 0.358‖ 1 ‖ 1 1 2 1 1‖ 0 ‖ 1 1 1 2 1‖ 0 ‖ 1 0 0 0 0‖ 0 ‖ 0 1 0 0 0
tj = t4I...
tj = t5I
MOA2 MOB2 NT0.505 0.345 0.150‖ 0 ‖ 1 1‖ 0 ‖ 0 0‖ 0 ‖ 0 0‖ 0 ‖ 0 0
α
tj = t5U...
1− α
α
tj = t4U...
1− α
1− α
1− α
αtj = t1U...
1− α
13
v0
...
13
v...
13
information content of different types of orders depend on an order’s direction, aggressiveness, and
on the prior order history?
The amount of adverse selection can change in two ways: The expected number of informed
traders can change, and the magnitude of asset value shocks can change. We consider four different
combinations of parameters with high and low informed-investor arrival probabilities (α = 0.8 and
0.2) and high and low value-shock volatilities (δ = 0.16 and 0.02). We call markets with δ = 0.02
low-volatility markets and markets with δ = 0.16 high-volatility markets, because the arriving
information is small relative to the tick size in the former parameterization and large relative to the
tick size in the later. In high-volatility markets, the final asset value v given good or bad news is
beyond the outside quotes A2 or B2, and so even market orders at the outside prices are profitable
for the informed traders. However, in low-volatility markets v will always be within the inside
quotes, and so market orders at A2 and B2 are not profitable for informed investors.
Figure 3: β Distribution and Upper Envelope for Informed Investor Iv at time t1.This figure shows the private-value factor β ∼ Tr[N (µ, σ2)] distribution superimposed on the plot of the expectedpayoffs the informed investor Iv with good news at time t1 for each equilibrium order type MOA2, MOB2, LOA2,LOA1, LOB1, LOB2, NT , (solid colored lines) when the book opens [0 0 0 0]. The dashed line shows the investor’supper envelope for the optimal orders. The vertical black lines show the β-thresholds at which two adjacent optimalstrategies yield the same expected payoffs. For example LOA1 is the optimal strategy for values of β between 0 andthe first vertical black line; LOA2 is instead the optimal strategy for the values of beta between the first and thesecond vertical lines. The parameters are α = 0.8, δ = 0.16, µ = 1, σ = 1.5, and κ = 0.10.
MOA2LOB1LOB2LOA2LOA1
0.5 1.0 1.5 2.0β
-0.4
-0.2
0.2
0.4
0.6
MOA2
MOB2
LOA2
LOA1
LOB1
LOB2
NT
18
2.1 Uninformed traders with random private-value motives
In our first model specification, only uninformed traders have random private-value factors. In-
formed traders have fixed neutral private-value factors β = 1. Thus, as in Kyle (1985), there is a
clear differentiation between investors who speculate on private information and those who trade
for purely non-informational reasons. Our model differs from Kyle (1985) in that informed and
uninformed investors can trade using both limit and market orders rather than being restricted to
market orders.
2.1.1 Trading strategies
We begin by investigating who supplies and takes liquidity and how these decisions change with
the amount of adverse selection. Table 1 reports results about trading early in the day at time t1
using a 2 × 2 format. Each of the four cells correspond to different combinations of parameters.
Comparing cells horizontally shows the effect of a change in the value-shock size δ while holding
the arrival probability α for informed traders fixed. Comparing cells vertically shows the effect of
a change in the informed-investor arrival probability while holding the value-shock size fixed. In
each cell corresponding to a set of parameter values, there are four columns reporting conditional
results for informed investors with good news, neutral news, and bad news about the asset (Iv, Iv0 ,
Iv) and for an uninformed investor (U) and a fifth column with the unconditional market results
(Uncond). The table reports the order-submission probabilities for the informed and uninformed in-
vestors and the corresponding unconditional order-arrival probabilities and several market-quality
metrics. Specifically, we report the expected bid-ask spread conditioning on the three informed
investor types, E[Spread |Iv], conditional on an uninformed trader E[Spread |U ], and also the un-
conditional market moment E[Spread] and the corresponding expected depths at the inside prices
(A1 and B1) and the total depths (A1 +A2 and B1 +B2) on each side of the market. In addition,
we report the probability-weighted contributions to the different investors’ gains-from-trade com-
ing from limit orders, market orders, and their total expected gains-from-trade. Table B1 in the
Numerical Appendix provides additional results about conditional and unconditional future execu-
tion probabilities for the different orders (PEX(xtj )) and also the uninformed investor’s updated
19
expected asset value E[v|xtj ] given different types of buy orders xt1 at time t1. Expectations given
sell orders are symmetric on the other side of E[v] = 1.
Table 2 shows average results for times t2 through t4 during the day using a similar 2×2 format.
The averages are across time and trading histories. Comparing results for time t1 with the trading
averages for t2 through t4 shows intraday changes in properties of the trading process. There is no
table for time t5, because only market orders are used at the market close.
20
Table 1: Trading Strategies, Liquidity, and Welfare at Time t1 in an Equilibrium with InformedTraders with β = 1 and Uninformed Traders with β ∼ Tr[N (µ, σ2)]. This table reports results fortwo different informed-investor arrival probabilities α (0.8 and 0.2) and two different value-shock volatilities δ (0.16and 0.02). The private-value factor parameters are µ = 1 and σ = 1.5, and the tick size is κ = 0.10. Each cellcorresponding to a set of parameters reports the equilibrium order-submission probabilities, the expected bid-askspreads and expected depths at the inside prices (A1 and B1) and total depths on each side of the market at timet1 as well as the welfare expectation of market participants. The first four columns in each parameter cell are forinformed traders with positive, neutral and negative signals, (Iv,Iv0 ,Iv
¯) and for uninformed traders (U). The fifth
column (Uncond.) reports unconditional results for the market.
δ = 0.16 δ = 0.02
Iv Iv0 Iv¯
U Uncond. Iv Iv0 Iv¯
U Uncond.
LOA2 0 0.500 0.650 0.143 0.335 0 0.500 1.000 0.052 0.410LOA1 0 0 0.350 0 0.093 0 0 0 0.079 0.016LOB1 0.350 0 0 0 0.093 0 0 0 0.079 0.016LOB2 0.650 0.500 0 0.143 0.335 1.000 0.500 0 0.052 0.410
MOA2 0 0 0 0.357 0.071 0 0 0 0.369 0.074MOA1 0 0 0 0 0 0 0 0 0 0MOB1 0 0 0 0 0 0 0 0 0 0MOB2 0 0 0 0.357 0.071 0 0 0 0.369 0.074NT 0 0 0 0 0 0 0 0 0 0
α = 0.8E[Spread |·] 0.265 0.300 0.265 0.300 0.281 0.300 0.300 0.300 0.284 0.297E[Depth A2+A1 |·] 1.000 1.500 2.000 1.143 1.429 1.000 1.500 2.000 1.131 1.426E[Depth A1 |·] 0 0 0.350 0 0.093 0 0 0 0.079 0.016E[Depth B1 |·] 0.350 0 0 0 0.093 0 0 0 0.079 0.016E[Depth B1+B2 |·] 2.000 1.500 1.000 1.143 1.429 2.000 1.500 1.000 1.131 1.426
E[Welfare LO |·] 0.034 0.053 0.034 0.018 0.029 0.069 0.029 0.015E[Welfare MO |·] 0 0 0 0.337 0 0 0 0.339E[Welfare |·] 0.034 0.053 0.034 0.355 0.029 0.069 0.029 0.354
LOA2 0 0.500 0.110 0.063 0.091 0 0.500 1.000 0.063 0.150LOA1 0 0 0.890 0.374 0.358 0 0 0 0.397 0.318LOB1 0.890 0 0 0.374 0.358 0 0 0 0.397 0.318LOB2 0.110 0.500 0 0.063 0.091 1.000 0.500 0 0.063 0.150
MOA2 0 0 0 0.064 0.051 0 0 0 0.040 0.032MOA1 0 0 0 0 0 0 0 0 0 0MOB1 0 0 0 0 0 0 0 0 0 0MOB2 0 0 0 0.064 0.051 0 0 0 0.040 0.032NT 0 0 0 0 0 0 0 0 0 0
α = 0.2E[Spread |·] 0.211 0.300 0.211 0.225 0.228 0.300 0.300 0.300 0.221 0.236E[Depth A2+A1 |·] 1.000 1.500 2.000 1.436 1.449 1.000 1.500 2.000 1.460 1.468E[Depth A1 |·] 0 0 0.890 0.374 0.358 0 0 0 0.397 0.318E[Depth B1 |·] 0.890 0 0 0.374 0.358 0 0 0 0.397 0.318E[Depth B1+B2 |·] 2.000 1.500 1.000 1.436 1.449 2.000 1.500 1.000 1.460 1.468
E[Welfare LO |·] 0.273 0.146 0.273 0.316 0.081 0.150 0.081 0.360E[Welfare MO |·] 0 0 0 0.099 0 0 0 0.064E[Welfare |·] 0.273 0.146 0.273 0.415 0.081 0.150 0.081 0.424
21
Table 2: Averages for Trading Strategies, Liquidity, and Welfare across Times t2 through t4 forInformed Traders with β = 1 and Uninformed Traders with β ∼ Tr[N (µ, σ2)]. This table reports resultsfor two different informed-investor arrival probabilities α (0.8 and 0.2) and for two different asset-value volatilitiesδ (0.16 and 0.02). The private-value factor parameters are µ = 1 and σ = 1.5, and the tick size is κ = 0.10. Eachcell corresponding to a set of parameters reports the equilibrium order-submission probabilities, the expected bid-askspreads and expected depths at the inside prices (A1 and B1) and total depths on each side of the market at timet1 as well as the welfare expectation of market participants. The first four columns in each parameter cell are forinformed traders with positive, neutral and negative signals, (Iv,Iv0 ,Iv
¯) and for uninformed traders (U). The fifth
column (Uncond.) reports unconditional results for the market.
δ = 0.16 δ = 0.02
Iv Iv0 Iv¯
U Uncond. Iv Iv0 Iv¯
U Uncond.
LOA2 0 0.191 0.051 0.157 0.096 0.399 0.255 0.108 0.026 0.209LOA1 0 0.258 0.257 0.023 0.142 0.192 0.239 0.288 0.064 0.205LOB1 0.257 0.258 0 0.023 0.142 0.288 0.239 0.192 0.064 0.205LOB2 0.051 0.191 0 0.157 0.096 0.108 0.255 0.399 0.026 0.209
MOA2 0.493 0 0 0.286 0.189 0 0 0 0.347 0.069MOA1 0.001 0 0 0.031 0.006 0 0 0 0.058 0.012MOB1 0 0 0.001 0.031 0.006 0 0 0 0.058 0.012MOB2 0 0 0.493 0.286 0.189 0 0 0 0.347 0.069NT 0.198 0.061 0.198 0.007 0.124 0.013 0.010 0.013 0.011 0.012
α = 0.8E[Spread |·] 0.217 0.212 0.217 0.251 0.223 0.227 0.228 0.227 0.278 0.237E[Depth A2+A1 |·] 1.047 2.276 2.480 1.755 1.899 2.165 2.300 2.433 1.608 2.161E[Depth A1 |·] 0 0.438 0.829 0.243 0.387 0.226 0.362 0.506 0.131 0.318
E[Depth B1 |·] 0.829 0.438 0 0.243 0.387 0.506 0.362 0.226 0.131 0.318E[Depth B1+B2 |·] 2.480 2.276 1.047 1.755 1.899 2.433 2.300 2.165 1.608 2.161E[Welfare LO |·] 0.010 0.020 0.010 0.106 0.014 0.013 0.014 0.005E[Welfare MO |·] 0.009 0 0.009 0.298 0 0 0 0.354E[Welfare |·] 0.019 0.020 0.019 0.405 0.014 0.013 0.014 0.359
LOA2 0 0.358 0.508 0.102 0.139 0.375 0.389 0.443 0.093 0.155LOA1 0 0.122 0.258 0.056 0.070 0.044 0.096 0.116 0.066 0.070LOB1 0.258 0.122 0 0.056 0.070 0.116 0.096 0.044 0.066 0.070LOB2 0.508 0.358 0 0.102 0.139 0.443 0.389 0.375 0.093 0.155
MOA2 0.130 0 0 0.219 0.184 0 0 0 0.218 0.175MOA1 0.088 0 0 0.119 0.101 0 0 0 0.120 0.096MOB1 0 0 0.088 0.119 0.101 0 0 0 0.120 0.096MOB2 0 0 0.130 0.219 0.184 0 0 0 0.218 0.175NT 0.016 0.035 0.016 0.006 0.010 0.022 0.030 0.022 0.005 0.009
α = 0.2E[Spread |·] 0.205 0.190 0.205 0.280 0.264 0.221 0.217 0.221 0.300 0.284E[Depth A2+A1 |·] 1.305 2.089 2.512 1.583 1.660 1.932 2.091 2.257 1.576 1.680E[Depth A1 |·] 0.194 0.451 0.740 0.301 0.333 0.346 0.414 0.442 0.262 0.290E[Depth B1 |·] 0.740 0.451 0.194 0.301 0.333 0.442 0.414 0.346 0.262 0.290E[Depth B1+B2 |·] 2.512 2.089 1.305 1.583 1.660 2.257 2.091 1.932 1.576 1.680
E[Welfare LO |·] 0.119 0.086 0.119 0.052 0.060 0.064 0.060 0.050E[Welfare MO |·] 0.018 0 0.018 0.343 0 0 0 0.342E[Welfare |·] 0.137 0.086 0.137 0.394 0.060 0.064 0.060 0.392
22
Result 1 Changes in adverse selection due to the value-shock size δ affect trading strategies
differently than changes in the informed-investor arrival probability α.
Consider the directionally informed investors Iv and Iv with good or bad news. First, hold
the informed-investor arrival probability α fixed and increase the amount of adverse selection by
increasing the value-shock volatility δ. In a low-volatility market in which value shocks ∆ are small
relative to the tick size, informed traders with good and bad news are unwilling to pay a large tick
size and instead act as liquidity providers who supply liquidity asymmetrically depending on the
direction of their information. This can be seen in Table 1 where in both of the two parameter
cells on the right (with α = 0.8 and 0.2 and a small δ = 0.02) informed investors Iv and Iv at time
t1 use limit orders at the outside quotes A2 and B2 exclusively. In contrast, in a high-volatility
market where value shocks are large relative to the tick size, informed investors with good or bad
news trade more aggressively. This can be seen in the left two parameterization cells (with α = 0.8
and 0.2 and a large δ = 0.16) where informed investors Iv and Iv use limit orders at both the inside
quotes A1 and B1 as well at the outside quotes with positive probability at time t1.
Now compare this to the effect of a change in the amount of adverse selection due to a change
in the informed-investor arrival probability α while holding the value-shock size δ fixed. In this
case, changing the amount of adverse selection does not affect which order informed investors with
good and bad news use at time t1. This can be seen by comparing the lower two parameter cells
(with δ = 0.02 and 0.16 and a small α) with the upper two parameter cells (with the same δs and
a larger α).
The average order-submission probabilities at times t2 through t4 in Table 2 are qualitatively
similar to those for time t1. When δ is small, informed investors with good and bad news tend
to supply liquidity via limit orders following strategies that are somewhat asymmetric on the two
sides of the market given the direction of their small amount of private information Iv and Iv. In
contrast, when the value-shock volatility δ is larger in a high-volatility market, informed investors
with good or bad news at times t2 to t4 switch from providing liquidity on both sides of the market
to using a mix of taking liquidity via market orders and supplying liquidity via limit orders on
the same side of the market as their information. Thus, once again, the trading strategies for
23
informed investors Iv and Iv are qualitatively similar holding δ fixed and changing α, but their
trading strategies change qualitatively when α is held fixed and δ is changed.
Next, consider informed investors I0 who know that the value shock ∆ is 0 and, thus, that
the unconditional prior v0 is correct. Tables 1 and 2 show that their liquidity provision trading
strategies are qualitatively the same at time t1 and on average over times t2 through t4. In constrast,
uninformed investors U become less willing to provide liquidity via limit orders at the inside quotes
as the adverse selection problem they face using limit orders worsens. Rather, they increasingly
take liquidity via market orders or supply liquidity by less aggressive limit orders at the outside
quotes. The reduction in liquidity provision at the inside quotes by uninformed investors is true at
time t1 (Table 1) and at times t2 through t4 (Table 2) both when the value shocks become larger
and when the arrival probability of informed investors increases.
In this context, there are two noteworthy equilibrium effects. First, while the uninformed U
investors reduce their liquidity provision at the inside quotes as adverse selection increases, the I0
informed investors increase their liquidity provision at the inside quotes. This is because I0 informed
investors have an advantage over the uninformed U investors in that there is no adverse selection
risk for them. These results are qualitatively consistent with the intuition of Bloomfield, O’Hara
and Saar (BOS, 2005). Informed traders are more likely to use limit orders than market orders
when the value-shock volatility is low (and, thus, the profitability from trading on information
asymmetries is low), and to use market orders when the reverse is true.
Second, uninformed U investors are unwilling to use aggressive limit orders at the inside quotes
when the adverse selection risk is sufficiently high as in the upper left parametrization (α = 0.8 and
δ = 0.16). This explains the fact that informed investors Iv and Iv use aggressive limit orders at
the inside quotes with a higher probability in the lower left parametrization (α = 0.2 and δ = 0.16)
than in the upper left parameterization. At first glance this might seems odd since competition
from future informed investors (and the possibility of being undercut by later limit orders) is greater
when the informed-investor arrival probability is large (α = 0.8) than when α is smaller. However,
in equilibrium there is camouflage from the uninformed U investors limit orders at the inside quotes
in the lower left parametization whereas in limit orders at the inside quotes are fully revealing in
24
the upper right parametrization.
2.1.2 Market quality
Market liquidity changes when the amount of adverse selection in a market changes. The standard
intuition, as in Kyle (1985), is that liquidity deteriorates given more adverse selection. For example,
Rosu (2016b) also finds worse liquidity (a wider bid-ask spread) given higher value volatility. How-
ever, we find that that is not always true.
Result 2 Liquidity need not always deteriorate when adverse selection increases.
Markets can become more liquid given greater value-shock volatility if, given the tick size, high
volatility makes the value shock ∆ large relative the price grid. In addition, different measures of
market liquidity — expected spreads, inside depth, and total depth — can respond differently to
changes in adverse selection.
The impact of adverse selection on market liquidity follows directly from the trading strategy
effects discussed above. Two intuitions are useful in understanding our market liquidity results.
First, different investors affect liquidity differently. Informed traders with neutral news (Iv0) are
natural liquidity providers. Thus, their impact on liquidity comes from whether they supply li-
quidity at the inside (A1 and B1) or outside (A2 and B2) prices. In contrast, informed traders
with directional news (Iv and Iv) and uninformed traders (U) can have a large impact on liquidity
depending on whether they opportunistically take or supply liquidity. Second, the most aggressive
way to trade (both on directional information and private values) is via market orders, which takes
liquidity. However, the next most aggressive way to trade is via limit orders at the inside prices.
Thus, changes in market conditions (i.e., δ and α) that make investors trade more aggressively (i.e.,
that reduce their use of limit orders at the outside prices, A2 and B2) can reduce or increase liquid-
ity depending on whether this stronger trading interest migrates to market orders or to aggressive
limit orders at the inside quotes, A1 and B1.
Our analysis shows that the standard intuition that adverse selection reduces market liquidity
depends on the relative magnitudes of asset value shocks and the tick size. In Table 1, the expected
25
spread narrows at time t1 (markets become more liquid) when the value-shock volatility δ increases
(comparing parameterizations horizontally so that α is kept fixed). Liquidity improves in these
cases because the informed traders Iv and Iv submit limit orders at the inside quotes in these high-
volatility markets, whereas they only use limit orders at the outside quotes in low-volatility markets.
In constrast, the expected spread at time t1 widens when the informed-investor arrival probability
α increases holding the value-shock size δ constant, as predicted by the standard intuition. The
evidence against the standard intuition is even stronger in Table 2. At times t2 through t4, the
expected spread narrows both when information becomes more volatile (δ is larger) and when there
are more informed traders (when α is larger). The qualitative results for the expected depth at the
inside quotes goes in the same direction as the results for the expected spread. This is because both
results are driven by limit-order submissions at the inside quotes. The results for adverse selection
and total depth at both the inside and outside quotes are mixed. For example, Table 1 shows that
total depth at time t1 increases when value-shock volatility δ increases when the informed-investor
arrival probability α is high (comparing horizontally the two parametrizations on the top), but
decreases in δ when the informed α is low. In contrast, average total depth at times t2 through t4
in Table 2 is decreasing in the value-shock volatility (comparing parameterizations horizontally).
This is opposite the effect on the inside depth. Thus, these different liquidity results are mixed.
The main result in this section is that the relation between adverse selection and market liquidity
is more subtle than the standard intuition. Increased adverse selection can improve liquidity.
The ability of investors to choose endogenously whether to supply or demand liquidity and at
what limit prices is what can overturn the standard intuition. The results from this specification
are comparable with Goettler et al. (2009). Goettler et al. (2009) have endogenous information
acquisition and therefore they have no regimes with both informed and uninformed traders having
an intrinsic motive to trade. However, they have a regime with informed traders having no private-
value trading motive and uninformed having only private-value motives. In this regime, when
volatility increases, informed traders reduce their provision of liquidity and increase their demand
of liquidity; with the opposite holding for uninformed traders. Our results are more nuanced.
Increased value-shock volatility is associated with increased liquidity supply in some cases and with
26
decreased liquidity in others. This is because the tick size of the price grid constrains the prices at
which liquidity can be supplied and demanded.
2.1.3 Information content of orders
Traders in real-world markets and empirical researchers are interested in the information content
of different types of arriving orders.10 A necessary condition for an order to be informative is that
informed investors use it. However, the magnitude of order informativeness is determined by the
mix of equilibrium probabilities with which both informed and uninformed traders use an order. If
uninformed traders use the same orders as informed investors, they add noise to the overall price
discovery process, and orders become less informative. In our model, the mix of information-based
and noise-based orders depends on the underlying proportion of informed investors α and and the
value-shock volatility δ.
We expect different market and limit orders to have different information content. A natural
conjecture is that the sign of the information revision associated with an order should agree with the
direction of the order (e.g, buy market and limit orders should lead to positive valuation revisions).
Another natural conjecture is that the magnitude of information revisions should be greater for
more aggressive orders. However, while the sign conjecture is true in our first model specification,
the order aggressiveness conjecture does not alway hold here.
Result 3 Order informativeness is not always increasing in the aggressiveness of an order.
This, at-first-glance surprising, result is another consequence of the impact of the tick size on how
informed investors trade on their information. As a result, the relative informativeness of different
market and limit orders can flip in high-volatility and low-volatility markets given a fixed tick size.
The result is immediate for market orders versus (less aggressive) limit orders in low-volatility
markets in which informed investors avoid market orders (see table 1). However, we show here that
it also can hold for aggressive limit orders at the inside quotes A1 and B1 versus less aggressive
limit orders at the outside quotes A2 and B2.
10Fleming et al. (2017) extend the VAR estimation approach of Hasbrouck (1991) to estimate the price impacts oflimit orders as well as market orders. See also Brogaard et al. (2016).
27
Figure 4: Informativeness of Orders after Trading at Time t1 for the Model with Informed Traders with β = 1 and UninformedTraders with β ∼ Tr[N (µ, σ2)]. This figure plots the Informativeness of the equilibrium orders at the end of t1 against the probability of orderexecution. We consider four different combinations of informed investors arrival probability. The informativeness of an order is measured as E[v|xt1 ]−E[v],where xt1 denotes one of the different possible orders that can arrive at time t1.
Figure 4 shows the informativeness of different types of orders at time t1. Informativeness at
time t1 is measured here as the revision E[v|xt1 ] − E[v] in the uninformed investor’s expectation
of the terminal value v after observing different types of orders xt1 at time t1. The informational
revisions for the different orders are plotted against the respective order-execution probabilities on
the horizontal axis. Orders with higher execution probabilities are statistically more aggressive than
orders with low execution probabilities. The results for the four parameterizations are indicated
using different symbols: high vs low informed-investor arrival probabilities (circles vs squares), and
high vs low value-shock volatility (large vs small symbols). These are described in the figure legend.
For example, in the low α and high δ scenario (large squares), the informativeness of a limit buy
order at B1 at time t1 is 0.026 and the order-execution probability is 78.9 percent (see Table B1 in
the Numerical Appendix).
Consider first the cases with high informed-investor arrival probabilities. The case with a high
informed-investor arrival probability and high value-shock volatility is denoted with large circles.
Informed investors in this case use limit orders at both the outside quotes (LOA2 and LOB2) and
inside quotes (LOA1 and LOB1) at time t1, so these are therefore the only informative orders.
Since uninformed investors also use the outside limit orders, they are not fully revealing, however
the inside limit orders are fully revealing. Thus, the price impacts for the inside and outside limit
orders here are consistent with the order aggressiveness conjecture. The market orders (MOB2
and MOA2) are also used in equilibrium, but only by uninformed investors (U). Thus, they are
not informative. While market orders would be profitable for the informed investors, the potential
price improvement with the limit orders leads informed investors to use the limit orders despite
the zero price impact and guaranteed execution probability of the market orders. Since both inside
and outside limit orders have larger price impacts than the market orders, this case is inconsistent
with the aggressiveness conjecture.
Next, consider the case of low value-shock volatility and high informed-investor arrival prob-
ability, denoted here with small circles. Once again, the order-aggressiveness conjecture is not
true. The most informative orders are now, not the most aggressive orders, but rather the most
patient limit orders posted at A2 and B2 (since these are the only orders used by informed in-
29
vestors). The market orders and more aggressive inside limit orders are non-informative here (since
only uninformed investors with extreme βs use them). In this case, this — again at first glance
perhaps counterintuitive — result is a consequence of the fact that the tick size is large relative
to the informed trader’s potential information. Low-volatility makes market orders unprofitable
for informed traders given good and bad news, and it increases the price improvement attainable
through limit orders deeper in the book relative to limit orders at the inside quotes.
Similar results hold when the proportion of insiders is low (α = 0.2). When the asset-value
volatility is high (large squares), the most aggressive orders (LOB1 and LOA1) are again the
most informative ones in contrast to the market orders. However, when volatility is low (small
squares), the most informative orders, as before, are the least aggressive orders (LOB2 and LOA2).
Therefore, the potential failures of the order-aggressiveness conjecture are robust to variation in
informed-investor arrival probabilities and value-shock volatility.
30
Figure 5: Informativeness of the Order History for the Model with Informed Traders with β = 1 andUninformed Traders with β ∼ Tr[N (µ, σ2)] for Times t3 and t4. This Figure shows the path-contingentBayesian value-forecast revision E(v|xtj , Ltj−1 ,Ltj−2)− E(v|xtj , Ltj−1), which shows the change in the uninformedtraders’ expected value of the fundamental for different histories, given the order xtj and the state of the book Ltj−1 .We only consider orders when they are equilibrium orders across the trading periods. The candlesticks indicate foreach of these two metrics the maximum, the minimum, the median and the 75th (and 25th) percentile respectivelyas the top (bottom) of the bar.
I) Parameters: α = 0.8, δ = 0.16
(a) LOB1
t3 t4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
(b) LOB2
t3 t4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
(c) MOA1
t3 t4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
(d) MOA2
t3 t4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
II) Parameters: α = 0.8, δ = 0.02
(a) LOB1
t3 t4
-0.02
-0.01
0.00
0.01
0.02
(b) LOB2
t3 t4
-0.02
-0.01
0.00
0.01
0.02
(c) MOA1
t3 t4
-0.02
-0.01
0.00
0.01
0.02
(d) MOA2
t3 t4
-0.02
-0.01
0.00
0.01
0.02
Figure 5: (Continued)
I) Parameters: α = 0.2, δ = 0.16
(a) LOB1
t3 t4
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
(b) LOB2
t3 t4
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
(c) MOA1
t3 t4
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
(d) MOA2
t3 t4
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
II) Parameters: α = 0.2, δ = 0.02
(a) LOB1
t3 t4
-0.010
-0.005
0.000
0.005
0.010
0.015
0.020
(b) LOB2
t3 t4
-0.010
-0.005
0.000
0.005
0.010
0.015
0.020
(c) MOA1
t3 t4
-0.010
-0.005
0.000
0.005
0.010
0.015
0.020
(d) MOA2
t3 t4
-0.010
-0.005
0.000
0.005
0.010
0.015
0.020
2.1.4 Non-Markovian learning
This section investigates the role of the order history on Bayesian learning. The first question we
consider is whether the prior order history has information about the value shock ∆ in excess of
the information in the current limit order book. The candlestick plots in Figure 5 show how the
information content of an arriving order xtj at time tj is affected by conditioning on the prior
order history Ltj−2 in addition to conditioning on the current book Ltj−1 . Each figure is for a
different combination of adverse-selection parameters. The horizontal axis shows the times t3 and
t4 at which different orders xtj are submitted. Note that times t1 and t2 are not included in
these plots. This is because the question studied here requires a time tj at which an order xtj
arrives, a time tj−1 from which there is an incoming current book Ltj−1 , and then at least one time
before tj−1 so that there can be a non-trivial history Ltj−2 . The vertical axis shows the Bayesian
revision E[v|xtj , Ltj−1,Ltj−2 ] − E[v|xtj , Ltj−1 ] in the uninformed investor’s expected asset value
conditional on different order history paths ending with an order xtj at time tj and book Ltj−1 at
time tj−1. In particular, note that different order-book pairs (xtj , Ltj−1) are preceded in equilibrium
by different histories Ltj−2 . If learning is Markov , then the prior order history Ltj−2 should convey
no additional information beyond that in the current book Ltj−1 . Each of the individual subplots
corresponds to a different order xtj at time tj . The number of subplots for a given parameterization
can vary depending on which orders are used in the different equilibria. The candlestick plots show
the maximum and minimum values, the interquartile range, and the median of the impact of the
prior history Ltj−2 on the valuation revision.
The main result from Figure 5 is that there is substantial informational variation in the Bayesian
revisions conditional on different trading histories. Thus, we have
Result 4 The price discovery dynamics can be significantly non-Markovian.
Given that learning is non-Markov, our next question is about how the size of the valuation
revisions depends on the prior trading history. In Figure 6, the horizontal axis is the price impact
of different orders at t1, and the vertical axis is the price impact of different equilibrium orders
at time t2 conditional on different order submissions at time t1. Consistently with our previous
33
Figure 6: Order Informativeness for the Model with Informed Traders with β = 1 and UninformedTraders with β ∼ Tr[N (µ, σ2)] for times t1 to t2 and parameters α = 0.8, δ = 0.16. The horizontalaxis reports E(v|xt1)−E(v) which shows how the uninformed traders’ Bayesian value-forecast changes with respectto the unconditional expected value of the fundamental when uninformed traders observe at t1 an equilibrium orderxt1 . The vertical axis reports E(v|xt2 , xt1)−E(v) which shows how the uninformed traders’ Bayesian value-forecastchanges with respect to the unconditional expected value of the fundamental when uninformed traders observe atxt2 at t2. We consider all the equilibrium strategies at t1 and t2 which are symmetrical. Red (green) circles showequilibrium sell (buy) orders at t2.
analysis, the size of the valuation revision crucially depends on the insiders’ equilibrium strategies.
As informed investors do not use market orders at t1, market orders do not have a price impact at
t1 which is also the reason why the price impact of any order at t2 conditional on a market order
at t1 lays on the vertical axis. Interestingly, there are no observations in the second and fourth
quadrants in our model, which means there are no sign reversals in the direction of the cumulative
price impacts. The first and third quadrants (which are perfectly symmetrical) show the duplets
of orders which have a positive and a negative price impact, respectively. The duplets with the
highest price impact are driven by the insiders’ equilibrium strategies at t1 and are limit orders
34
at the inside quotes followed any other order. In fact, Table 1 shows that insiders’ limit orders at
the inside quotes at t1 are fully revealing. So once more the price impact does not depend on the
aggressiveness of the orders but on the informed investors’ orders choice. Overall, Figure 6 also
confirms that the price impact is non-Markovian: for example the price impact of MOB2 at t2 may
be either positive or negative depending on whether it is preceded by LOB2 or LOA2 at t1.
2.1.5 Summary
The analysis our of first model specification has identified a number of empirically testable predic-
tions. First, liquidity and the relative information content of different orders differ in high-volatility
markets in which value shocks are large relative to the tick size vs. in low-volatility markets in
which value shocks are small relative to the tick size. Second, the price impact of order flow should
vary conditional on different trading histories and the current book at the time new orders are
submitted.
2.2 Informed and uninformed traders where both have private-value motives
Our second model specification generalizes our earlier analysis so that now informed investors
also have random private-valuation factors β with the same truncated Normal distribution β ∼
Tr[N (µ, σ2)] as the uninformed investors. Hence, informed traders not only speculate on their
information, but they also have a private-value motive to trade. As a result, informed investors
with the same signal may end up buying and selling from each other. We use this second model
specification to show the robustness of the results in Section 2.1.
2.2.1 Trading strategies
Tables 3 and 4 report numerical results for our second model specification for time t1 by itself
and for averages over times t2 through t4. Since all investors have private-value motives to trade,
we see that now all investors use all of the possible limit orders at time t1 and that directionally
informed and uninformed investors also use market orders. Over times t2 through t4, all investors
again use all types of limit orders and also market orders. In particular, directionally informed
35
Table 3: Trading Strategies, Liquidity, and Welfare at Time t1 in an Equilibrium with Informed andUninformed Traders both with β ∼ Tr[N (µ, σ2)]. This table reports results for two different informed-investorarrival probabilities α (0.8 and 0.2) and two different value-shock volatilities δ (0.16 and 0.02). The private-valuefactor parameters are µ = 1 and σ = 1.5, and the tick size is κ = 0.10. Each cell corresponding to a set of parametersreports the equilibrium order-submission probabilities, the expected bid-ask spreads and expected depths at theinside prices (A1 and B1) and total depths on each side of the market at time t1 as well as the welfare expectationof market participants. The first four columns in each parameter cell are for informed traders with positive, neutraland negative signals, (Iv,Iv0 ,Iv
¯) and for uninformed traders (U). The fifth column (Uncond.) reports unconditional
results for the market.
δ = 0.16 δ = 0.02
Iv Iv0 Iv¯
U Uncond. Iv Iv0 Iv¯
U Uncond.
LOA2 0.118 0.054 0.031 0.064 0.067 0.054 0.048 0.042 0.048 0.048LOA1 0.314 0.446 0.282 0.426 0.363 0.438 0.452 0.466 0.452 0.452LOB1 0.282 0.446 0.314 0.426 0.363 0.466 0.452 0.438 0.452 0.452LOB2 0.031 0.054 0.118 0.064 0.067 0.042 0.048 0.054 0.048 0.048
MOA2 0.256 0 0 0.009 0.070 0 0 0 0 0MOA1 0 0 0 0 0 0 0 0 0 0MOB1 0 0 0 0 0 0 0 0 0 0MOB2 0 0 0.256 0.009 0.070 0 0 0 0 0NT 0 0 0 0 0 0 0 0 0 0
α = 0.8E[Spread |·] 0.240 0.211 0.240 0.215 0.227 0.210 0.210 0.210 0.210 0.210E[Depth A2+A1 |·] 1.432 1.500 1.312 1.491 1.430 1.492 1.500 1.508 1.500 1.500E[Depth A1 |·] 0.314 0.446 0.282 0.426 0.363 0.438 0.452 0.466 0.452 0.452E[Depth B1 |·] 0.282 0.446 0.314 0.426 0.363 0.466 0.452 0.438 0.452 0.452E[Depth B1+B2 |·] 1.312 1.500 1.432 1.491 1.430 1.508 1.500 1.492 1.500 1.500
E[Welfare LO |·] 0.259 0.445 0.259 0.410 0.446 0.446 0.446 0.446E[Welfare MO |·] 0.187 0 0.187 0.015 0 0 0 0E[Welfare |·] 0.446 0.445 0.446 0.425 0.446 0.446 0.446 0.446
LOA2 0.063 0.051 0.042 0.051 0.051 0.049 0.048 0.046 0.048 0.048LOA1 0.356 0.449 0.476 0.449 0.445 0.441 0.452 0.464 0.452 0.452LOB1 0.476 0.449 0.356 0.449 0.445 0.464 0.452 0.441 0.452 0.452LOB2 0.042 0.051 0.063 0.051 0.051 0.046 0.048 0.049 0.048 0.048
MOA2 0.063 0 0 0 0.004 0 0 0 0 0MOA1 0 0 0 0 0 0 0 0 0 0MOB1 0 0 0 0 0 0 0 0 0 0MOB2 0 0 0.063 0 0.004 0 0 0 0 0NT 0 0 0 0 0 0 0 0 0 0
α = 0.2E[Spread |·] 0.217 0.210 0.217 0.210 0.211 0.210 0.210 0.210 0.210 0.210E[Depth A2+A1 |·] 1.419 1.500 1.518 1.500 1.496 1.490 1.500 1.510 1.500 1.500E[Depth A1 |·] 0.356 0.449 0.476 0.449 0.445 0.441 0.452 0.464 0.452 0.452E[Depth B1 |·] 0.476 0.449 0.356 0.449 0.445 0.464 0.452 0.441 0.452 0.452E[Depth B1+B2 |·] 1.518 1.500 1.419 1.500 1.496 1.510 1.500 1.490 1.500 1.500
E[Welfare LO |·] 0.394 0.445 0.394 0.442 0.447 0.446 0.447 0.446E[Welfare MO |·] 0.059 0 0.059 0 0 0 0 0E[Welfare |·] 0.453 0.445 0.453 0.442 0.447 0.446 0.447 0.446
36
Table 4: Averages for Trading Strategies, Liquidity, and Welfare across Times t2 through t4 forInformed and Uninformed Traders both with β ∼ Tr[N (µ, σ2)]. This table reports results for two differentinformed-investor arrival probabilities α (0.8 and 0.2) and for two different asset-value volatilities δ (0.16 and 0.02).The private-value factor parameters are µ = 1 and σ = 1.5, and the tick size is κ = 0.10. Each cell corresponding to aset of parameters reports the equilibrium order-submission probabilities, the expected bid-ask spreads and expecteddepths at the inside prices (A1 and B1) and total depths on each side of the market at time t1 as well as thewelfare expectation of market participants. The first four columns in each parameter cell are for informed traderswith positive, neutral and negative signals, (Iv,Iv0 ,Iv
¯) and for uninformed traders (U). The fifth column (Uncond.)
reports unconditional results for the market.
δ = 0.16 δ = 0.02
Iv Iv0 Iv¯
U Uncond. Iv Iv0 Iv¯
U Uncond.
LOA2 0.140 0.121 0.090 0.114 0.117 0.127 0.123 0.119 0.123 0.123LOA1 0.108 0.058 0.050 0.067 0.071 0.057 0.053 0.048 0.053 0.053LOB1 0.050 0.058 0.108 0.067 0.071 0.048 0.053 0.057 0.053 0.053LOB2 0.090 0.121 0.140 0.114 0.117 0.119 0.123 0.127 0.123 0.123
MOA2 0.275 0.192 0.113 0.195 0.194 0.207 0.194 0.181 0.194 0.194MOA1 0.158 0.127 0.062 0.122 0.117 0.133 0.128 0.124 0.129 0.128MOB1 0.062 0.127 0.158 0.122 0.117 0.124 0.128 0.133 0.129 0.128MOB2 0.113 0.192 0.275 0.195 0.194 0.181 0.194 0.207 0.194 0.194NT 0.003 0.003 0.003 0.005 0.004 0.004 0.003 0.004 0.004 0.004
α = 0.8E[Spread |·] 0.253 0.259 0.253 0.274 0.259 0.268 0.269 0.268 0.269 0.268E[Depth A2+A1 |·] 1.599 1.600 1.537 1.563 1.576 1.590 1.593 1.596 1.593 1.593E[Depth A1 |·] 0.301 0.339 0.338 0.314 0.324 0.324 0.333 0.344 0.333 0.334E[Depth B1 |·] 0.338 0.339 0.301 0.314 0.324 0.344 0.333 0.324 0.333 0.334E[Depth B1+B2 |·] 1.537 1.600 1.599 1.563 1.576 1.596 1.593 1.590 1.593 1.593
E[Welfare LO |·] 0.089 0.071 0.089 0.072 0.067 0.067 0.067 0.067E[Welfare MO |·] 0.328 0.332 0.328 0.331 0.336 0.336 0.336 0.336E[Welfare |·] 0.418 0.403 0.418 0.404 0.403 0.403 0.403 0.403
LOA2 0.131 0.123 0.114 0.122 0.122 0.124 0.123 0.122 0.123 0.123LOA1 0.059 0.054 0.049 0.053 0.054 0.053 0.053 0.052 0.053 0.053LOB1 0.049 0.054 0.059 0.053 0.054 0.052 0.053 0.053 0.053 0.053LOB2 0.114 0.123 0.131 0.122 0.122 0.122 0.123 0.124 0.123 0.123
MOA2 0.257 0.194 0.137 0.196 0.196 0.202 0.194 0.186 0.194 0.194MOA1 0.160 0.127 0.090 0.127 0.127 0.133 0.128 0.124 0.128 0.128MOB1 0.090 0.127 0.160 0.127 0.127 0.124 0.128 0.133 0.128 0.128MOB2 0.137 0.194 0.257 0.196 0.196 0.186 0.194 0.202 0.194 0.194NT 0.004 0.003 0.004 0.004 0.004 0.004 0.003 0.004 0.004 0.004
α = 0.2E[Spread |·] 0.266 0.267 0.266 0.269 0.269 0.269 0.269 0.269 0.269 0.269E[Depth A2+A1 |·] 1.547 1.595 1.636 1.591 1.591 1.587 1.593 1.599 1.592 1.592E[Depth A1 |·] 0.288 0.334 0.378 0.332 0.332 0.327 0.333 0.339 0.333 0.333E[Depth B1 |·] 0.378 0.334 0.288 0.332 0.332 0.339 0.333 0.327 0.333 0.333E[Depth B1+B2 |·] 1.636 1.595 1.547 1.591 1.591 1.599 1.593 1.587 1.592 1.592
E[Welfare LO |·] 0.068 0.068 0.068 0.067 0.067 0.067 0.067 0.067E[Welfare MO |·] 0.348 0.334 0.348 0.335 0.336 0.336 0.336 0.336E[Welfare |·] 0.416 0.403 0.416 0.402 0.403 0.403 0.403 0.403
37
investors trade with and also sometimes opposite their asset-value information because their non-
informational private-value motive adds noise to their orders. Informed investor with neutral news
Iv0 no longer just provide liquidity using limit orders. Now, due to their private-value motive, they
sometimes also take liquidity using market orders.
Consider next the impact of the amount of adverse selection on trading behavior. Tables 3
and 4 show for time t1 and for trading averages over t2 through t4 respectively that the effects of
an increase in value-shock volatility on the strategies of informed traders with good or bad news
differs if we consider traders’ own or opposite side of the market. In particular, the “own” side of
the market for an informed investor with good news is the bid (buy) side of the limit order book.
The effect on the informed trader’s own-side behavior is similar to the previous model specification
in Section 2.1. With higher value-shock volatility, the private information about the asset value
is more valuable, and both Iv and Iv¯
investors change some of their aggressive limit orders into
market orders. Table 3 shows that, at time t1 when α = 0.8, the Iv investors reduce the strategy
probability for LOB1 orders from 0.466 to 0.282 and increase the strategy probability for MOA2
orders from 0 to 0.256, and symmetrically Iv¯
investors shifts from LOA1 to MOB2.
The effects of higher volatility on uninformed traders slightly differs if we consider t1 as opposed
to times t2 through t4. At t1 uninformed traders post slightly more aggressive orders when they
demand liquidity (the strategy probabilities for MOA2 and MOB2 increase from 0 to 0.009), and
more patient orders when they supply liquidity (the strategy probabilities for LOB2 and LOA2
increase slightly from 0.048 to 0.064). This change in order-submission strategies is the consequence
of uninformed traders now perceiving higher adverse selection costs. They feel safer hitting the
trading crowd at A2 and B2 and offering liquidity at more profitable price levels to make up for the
increased adverse selection costs. In later periods t1 through t4, as uninformed traders learn about
the fundamental value of the asset, they still take liquidity at the outside quotes (the probabilities
of MOA2 and MOB2 increase slightly to 0.195 in Table 4), but move to the inside quotes to supply
liquidity (LOA1 and LOB1 increase to 0.067 for times t2 through t4). As they learn about the
future value of the asset, uninformed traders perceive less adverse selection costs and can afford to
offer liquidity at more aggressive quotes. In contrast, the effects of increased value-shock volatility
38
on the trading behavior of Iv0 investors are relatively modest both at time t1 and at times t2
through t4.
The effects of an increase in volatility on the opposite side is different than on the own side. For
example, when asset-value volatility δ increases from 0.02 to 0.16, Iv investors at time t1 switch
on the own side from LOB1 limit orders to aggressive MOA2 market orders and at the same time
they switch on the opposite side from aggressive limit orders to more patient limit orders. The
reason why Iv investors with low private-values become more patient when selling via limit orders
on the opposite side is that they know that the execution probability of limit sells at A2 is higher
because other Iv investors in future periods will hit limit sell orders at A2 more aggressively given
that v is much bigger (see the increased order submission probabilities for MOA2 in Table 4).
2.2.2 Market quality
The effect of value-shock volatility on market liquidity is mixed in our second model specification.
This is not surprising given the nuanced effect of increased volatility on investor trading behavior,
particularly on informed trading behavior on the own and opposite sides of the market. At time
t1, holding the informed-investor arrival probability α fixed, increased value-shock volatility leads
to wider spreads and less total depth. However, the average effects over times t2 through t4 is
the opposite with increased asset-value volatility leading to narrower spreads and smaller depth.
This is due — in particular in the high α framework — to uninformed traders perceiving greater
adverse selction costs and therefore being less willing to supply liquidity. Interestingly, the effects
of an increase in the proportion of informed investors (α) on the equilibrium strategies of market
participants is qualitatively similar to that of an increase in volatility (δ) in this model.
Lastly, our model shows how an increase in volatility and in the proportion of insiders affect
the welfare of market participants. When volatility increases, directional informed investors are
generally better off as their signal is stronger and hence more profitable: At t1 their welfare is
unchanged with high proportion of insiders (0.446), whereas it increases in all the other scenarios,
with low proportion of insiders (0.453) and in later periods with both high and low α (0.418 and
0.416). At t1 uninformed traders are worse off because liquidity deteriorates with higher volatility.
39
At later periods the result is ambiguous: there are cases in which the uninformed investors are
better off and cases in which they are worse off.
2.2.3 Information content of orders
Figure 7 plots the Bayesian revisions for different orders at time t1 against the corresponding order-
execution probabilities for our second model specification. Once again, the magnitudes and signs
of the Bayesian updates depends on the mix of informed and uninformed investors who submit
these orders. Consider, for example, the market with both high value-shock volatility and a high
informed-investor arrival probability (large circles). The most informative orders are the market
orders MOA2 and MOB2 as they are chosen much more often by informed investors than by
uninformed investors. However, the next most aggressive orders are the inside limit orders LOB1
and LOA1, and they are less informative than the LOB2 and LOA2 limit orders. Even though
the aggressive limit orders LOB1 and LOA1 are posted with a relatively high probability (0.282
and 0.314) by informed investors Iv and Iv, they are also submitted with a high probability by
uninformed investors (0.426), and an even higher submission probability by Iv0 investors (0.446).11
As a result, they are less informative.12 Thus, this is another example in which order informativeness
is not increasing in order aggressiveness.
11Investor Iv¯
choose LOB1 and LOA1 with probability 0.314 and 0.282 respectively.12The informativeness of LOA1 and LOB1 in Table 3 are 0.004 and −0.004 respectively, whereas the informativeness
of LOA2 and LOB2 are 0.056 and −0.056 respectively.
40
Figure 7: Informativeness of Orders at the End of t1 for the Model with Informed and Uninformed Traders both with β ∼Tr[N (µ, σ2)]. This figure plots the Informativeness of the equilibrium orders at the end of t1 against the probability of execution. We consider fourdifferent combinations of informed investors arrival probability. The informativeness of an order is measured as E[v|xt1 ]− E[v], where xt1denotes one ofthe different possible orders that can arrive at time t1.
MOA2 δ 0.16,α 0.2
MOB2 δ 0.16,α 0.2
LOA2 δ 0.16,α 0.2
LOA1 δ 0.16,α 0.2
LOB1 δ 0.16,α 0.2
LOB2 δ 0.16,α 0.2
LOA2 δ 0.02,α 0.2
LOA1 δ 0.02,α 0.2
LOB1 δ 0.02,α 0.2
LOB2 δ 0.02,α 0.2
MOA2 δ 0.16,α 0.8
LOA2 δ 0.16,α 0.8
LOA1 δ 0.16,α 0.8
LOB1 δ 0.16,α 0.8
LOB2 δ 0.16,α 0.8
MOB2 δ 0.16,α 0.8
LOA1 δ 0.02,α 0.8
LOB1 δ 0.02,α 0.8
LOA2 δ 0.02,α 0.8
LOB2 δ 0.02,α 0.80.2 0.4 0.6 0.8 1.0Probability
-0.15
-0.10
-0.05
0.05
0.10
0.15
Informativeness
δ=0.02,α=0.2
δ=0.02,α=0.8
δ=0.16,α=0.2
δ=0.16,α=0.8
Perhaps even more surprisingly, the order-sign conjecture about order informativeness does not
always hold in our second specification. That is to say, the direction of orders is sometimes different
from the sign of their information content. For example, a limit sell LOA1 is informative of good
news (rather than bad news as one might expect) because limit sells at A1 are used by informed
investor to trade on the opposite side of their information (i.e., due to their private-value β factors)
more frequently than these orders are used to trade on the same side of their information. In
particular, Iv investors usually sell using market orders at MOB2 rather than using limit sells.
This goes back to our previous discussion of how informed investors trade differently on the own
side of their information (when their private value β reinforces the trading direction from their
information) and on the opposite side of their information (when their β reverses the trading
incentive from their information).
42
Figure 8: History Informativeness for Informed and Uninformed Traders both withβ ∼ Tr[N (µ, σ2)] for times t3 and t4. This Figure shows the path-contingent Bayesian value-forecastrevision E(v|xtj , Ltj−1 ,Ltj−2)− E(v|xtj , Ltj−1), which shows the change in the uninformed traders’ expected valueof the fundamental for different histories, given the order xtj and the state of the book Ltj−1 . We only consider orderswhen they are equilibrium orders across the trading periods. The candlesticks indicate for each of these two metricsthe maximum, the minimum, the median and the 75th (and 25th) percentile respectively as the top (bottom) of the bar.
I) Parameters: α = 0.8, δ = 0.16
(a) LOB1
t3 t4
-0.2
-0.1
0.0
0.1
0.2
(b) LOB2
t3 t4
-0.2
-0.1
0.0
0.1
0.2
(c) MOA1
t3 t4
-0.2
-0.1
0.0
0.1
0.2
(d) MOA2
t3 t4
-0.2
-0.1
0.0
0.1
0.2
II) Parameters: α = 0.8, δ = 0.02
(a) LOB1
t3 t4
-0.015
-0.010
-0.005
0.000
0.005
0.010
0.015
(b) LOB2
t3 t4
-0.015
-0.010
-0.005
0.000
0.005
0.010
0.015
(c) MOA1
t3 t4
-0.015
-0.010
-0.005
0.000
0.005
0.010
0.015
(d) MOA2
t3 t4
-0.015
-0.010
-0.005
0.000
0.005
0.010
0.015
Figure 8: (Continued)
I) Parameters: α = 0.2, δ = 0.16
(a) LOB1
t3 t4
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
(b) LOB2
t3 t4
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
(c) MOA1
t3 t4
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
(d) MOA2
t3 t4
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
II) Parameters: α = 0.2, δ = 0.02
(a) LOB1
t3 t4
-0.0015
-0.0010
-0.0005
0.0000
0.0005
0.0010
0.0015
(b) LOB2
t3 t4
-0.0015
-0.0010
-0.0005
0.0000
0.0005
0.0010
0.0015
(c) MOA1
t3 t4
-0.0015
-0.0010
-0.0005
0.0000
0.0005
0.0010
0.0015
(d) MOA2
t3 t4
-0.0015
-0.0010
-0.0005
0.0000
0.0005
0.0010
0.0015
Figure 9: Informativeness of the Equilibrium States of the Book: E[v|Book].This figure shows can-dlestick plots for for two measures of informativeness of a book across the periods t1 through t4. The first measure(plots a,b,c and d) is the uninformed traders’ expectation of the fundamental value of the asset (E[v|Book]) basedof the observation of the books across the periods t1 through t4. The second measure (plots e,f,g and h) is the path-contingent absolute forecast error πu
tj |v−E(v|Ltj−1 , xtj )|+π0tj |v0−E(v|Ltj−1 , xtj )|+πd
tj |v−E(v|Ltj−1 , xtj )| basedof the observation of the books across the periods t1 through t4. In the first column (plots a, c, e and g) we report themeasure of informativeness under the regime with high proportion of informed traders (α = 0.8) and high volatility(δ = 0.016), whereas in the second column (plots b, d, f and h) we report the same measure conditional on the samebooks but under the regime with low proportion of informed traders (α = 0.2) and high volatility (δ = 0.016). Weconsider all the possible equilibrium paths that lead again to the equilibrium states of the book and we report thesenumbers in the plot. As the number of periods increases also the number of E[v|Book] increases. The candlestickindicate the maximum, the minimum, the mean and the 75th (and 25th) percentile respectively as the top (bottom)of the bar and the red segment within the bar.
(a) Book 1001 α = 0.8, δ = 0.16 (b) Book 1001 α = 0.2, δ = 0.16
(c) Book 1101 α = 0.8, δ = 0.16 (d) Book 1101 α = 0.2, δ = 0.16
(e) Book 1001 α = 0.8, δ = 0.16
t1 t2 t3 t4
0.00
0.05
0.10
0.15
0.20
0.25
0.30
(f) Book 1001 α = 0.2, δ = 0.16
t1 t2 t3 t4
0.00
0.05
0.10
0.15
0.20
0.25
0.30
(g) Book 1101 α = 0.8, δ = 0.16
t1 t2 t3 t4
0.00
0.05
0.10
0.15
0.20
0.25
0.30
(h) Book 1101 α = 0.2, δ = 0.16
t1 t2 t3 t4
0.00
0.05
0.10
0.15
0.20
0.25
0.30
2.2.4 Non-Markovian price discovery
This section continues our investigation of the importance of non-Markovian effects in informa-
tion aggregation. Figure 8 shows once again that the Bayesian revisions E[v|xtj , Ltj−1 ,Ltj−2 ] −
E[v|xtj , Ltj−1 ] vary depending on the prior order history Ltj−2 . The plots here confirm our earlier
results about non-Markovian learning.
Figure 9 looks at the informativeness of history slightly differently. The top four plots (a, b, c,
and d) in the figure show the distributions of uninformed investors’ value expectations E[v|Book]
conditional on different paths ending with the indicated particular book. The fact that this ex-
pectation differs across different paths means that the value expectations are not Markovian. In
particular these expectations depend on the prior trading history. The bottom four plots (e,
f, g, and h) in Figure 9 show the path-dependent conditional expected absolute forecast error
E[|v − E(v|Ltj−1 , LOA1,tj )|] conditional on a limit order LOA1 at time tj . This is used here as a
measure of pricing accuracy. It indicates the degree of uninformed traders’ valuation forecast-error
dispersion conditional on the observed history of equilibrium orders ending with that specific order.
As time passes, the number of paths and, thus, the number of path-contingent forecasts increases
relative to t1.
Goettler et al. (2009) and Rosu (2016b) assume that information dynamics are Markovian and
that the current limit order book is a sufficient statistic for the information content of the prior
trading history. Figure 9 shows the uninformed investor’s expectation of the asset value conditional
on the path and various books. It also shows the expectation of these expectations across the paths,
which, by iterated expectation, is the expectation conditional on the book. Again, we see that the
trading history has substantial information content above and beyond the information in the book
alone. The figure also shows the standard deviation of the valuation forecast errors. Here again,
the results are non-Markovian.
2.3 Summary
The results for our second model specification — with a richer specification of the informed investors’
trading motives — confirm and extent the analysis from our first model specification. The main
46
findings are
• When all market participants trade not only to speculate on their signal but also to satisfy
their private-value motive, all investors use both market and limit orders in equilibrium.
• Increased value-shock volatility and an increased informed-investor arrival probability can
affect informed investor trading behavior differently when they trade with their information
or (because of private-value shocks) against their information.
• The effect of asset-value volatility and informed investor arrival probability on market liquidity
is mixed.
• The informativeness of an order can again be opposite the order direction and aggressiveness.
• The information content of order arrivals is history-dependent.
• Both order informativeness and the dispersion of believes increase with volatility and the
proportion of insiders. With higher volatility the insiders’ signal becomes stronger, whereas
with a higher proportion of insiders uninformed traders have more opportunities to learn.
3 Robustness
Our analysis makes a number of simplifying assumptions for tractability, but we conjecture that our
qualitative results are robust to relaxing these assumptions. We consider two of these assumptions
here. First, our model of the trading day only has five periods. Relatedly, our analysis abstracts
from limit orders being carried over from one day to the next. However, our results about the impact
of adverse selection on investor trading strategies and about order informativeness are driven in
large part by the relative size of information shocks and the tick size rather than by the number
of rounds of trading. In addition, increasing the trading horizon just leads to longer histories that
are potentially even more informative. Second, arriving investors are only allowed to submit single
orders that cannot be cancelled or modified subsequently. However, it seems likely that order flow
histories will still be informative if orders at different points in time are correlated due to correlated
actions of returning investors.
47
4 Conclusions
This paper has studied the information aggregation and liquidity provision processes in dynamic
limit order markets. We show a number of interesting theoretical properties in our model. First, in-
formed investors switch between endogenously demanding liquidity via market orders and supplying
liquidity via limit orders. Second, the information content/price impact of orders is non-monotone
in the direction of the order and in the aggressiveness of their orders. Third, the information ag-
gregation process is non-Markovian. In particular, the prior trading history has information content
beyond that in the current limit order book. We also show that the price impact of orders depends
on the prior trading history. In other words, a given order may have a very different price impact
following one trading history and another.
Our model suggests several interesting directions for future research. First, the model can
be enriched by allowing investors to trade dynamically over time (rather than just submitting an
order one time). In addition, if traders had a quantity decision, they might want to use multiple
orders. Second, the model could be extended to allow for trading in multiple co-existing limit order
markets. This would be a realistic representation of current equity trading in the US. Third, the
model could be used to study high frequency trading and the effect of different investors being able
to process and trade on different types of information at different latencies.
5 Appendix A: Algorithm for computing equilibrium
The computational problem to solve for a Perfect Bayesian Nash equilibrium in our model is
complex. Given investors’ equilibrium beliefs, the optimal order-submission problems in (5) and
(6) require computing limit-order execution probabilities and stock-value expectations conditional
on both the past trading history and on future state-contingent limit-order execution at each time tj
at each node of the trading game. For an informed trader (who knows the future value of the asset),
there is no uncertainty about the payoff of a market order. However, the payoff of a market order for
an uninformed trader entails uncertainty about the future asset value and therefore computing the
optimal order requires computing the expected stock value conditional on the prior trading history
48
up to time tj . For limit orders, the expected payoff depends on the future execution probability of
that limit order, which depends, in turn, on the optimal order-submission probabilities for future
informed and uninformed traders. In addition, the uninformed investors have a learning problem.
They extract information about the expected future stock value from both the past trading history
and also from state-contingent future order execution given that the future states in which limit
orders are executed are correlated with the stock value. Thus, optimal actions at each date t depend
on past and future actions where future actions also depend on the prior histories at future dates
(which included the action at date t) as traders dynamically update their equilibrium beliefs as the
trading process unfolds. In addition, rational expectations involves finding a fixed point so that
the equilibrium beliefs underlying the optimal order-submission strategies are consistent with the
execution probabilities and value expectations that those optimal strategies produce in equilibrium.
Our numerical algorithm uses backwards induction to solve for optimal order strategies given
a set of asset-value beliefs for all dates and nodes in the trading game and an iterative recursion
to solve for RE asset-value beliefs. The backwards induction makes order-execution probabilities
consistent with optimal future behavior by later arriving investors. It also takes future state-
contingent execution into account in the uninformed investors’ learning problem. We start at time
t5 — when traders only use market orders which allows us to compute the execution probabilities
of limit orders at t4 — and recursively solve the model for optimal trading strategies back to time
t1. We then embed the optimal order strategy calculation in an iterative recursion to solve for a
fixed point for the RE asset-value beliefs. In this recursion, the asset-value probabilities πv,r−1t from
round r− 1 are used iteratively as the asset-value beliefs in round r. In particular, these beliefs are
used in the learning problem of the uninformed investor to extract information about the ending
stock value v from the prior trading histories. They also affect the behavior of informed investors
whose order-execution probability beliefs depend in part on the behavior of uninformed traders.
We iterate this recursion to find a RE fixed point in investor beliefs.
In a generic round r of our recursion, investors’ asset-value beliefs are set to be the asset-value
probabilities from the previous recursive round r− 1. In particular, at each time tj in each node of
the trading process, the round r−1 probabilities are used as priors in computing traders’ conditional
49
payoffs in round r when computing expected order payoffs and optimal orders:
maxx∈Xtj
ϕI, r(x | v,Ltj−1) = [β v0 + ∆− p(x)]Prr−1(θxtj | v,Ltj−1) (12)
and
maxx∈Xtj
ϕU, r(x |Ltj−1) = [β v0 + Er−1[∆ |Ltj−1 , θxtj ]− p(x)]Prr−1(θxtj |Ltj−1) (13)
where
Er−1[∆|Ltj−1 , θxtj ] = (πv, r−1
tjv + πv0, r−1
tjv0 + π
v, r−1tj
v)− v0 (14)
πv, r−1tj
=Prr−1(θxtj |v,Ltj )
Prr−1(θxtj |Ltj )πv, r−1tj
(15)
The resulting order-submission strategies xtj ,r in round r are then used to to compute new asset-
value asset value beliefs for the next recursive round r + 1.
The recursion is started in round r = 1 by setting the beliefs of uninformed traders about the
fundamental value of the asset at each time tj in the backwards induction to be the unconditional
priors given in (1). In particular, the algorithm starts by ignoring conditioning on history in the
initial round r = 1. Hence traders’ expected payoffs on an order x in round r = 1 simplify to:
maxx∈Xtj
ϕUr=1(x |Ltj−1) = [β v0 + E[∆]− p(x)]Pr(θxtj ) (16)
maxx∈Xtj
ϕIr=1(x | v,Ltj−1) = [β v0 + ∆− p(x)]Pr(θxtj | v) (17)
In each round r given the asset-value beliefs in that round, we solve for investors’ optimal
trading strategies by backward induction. Starting at t5, the execution probability of new limit
orders is zero, and therefore optimal order-submission strategies only use market orders. Given the
linearity of the expected payoffs in the private-value factor β (equations (16) and (17)), the optimal
50
trading strategies for an informed trader at t5 are13
xt5,I,r(β|Lt4 , v) =
MOBi,t5 if β ∈ [0, β
MOBi,t5,NT
t5,I,r)
NT if β ∈ [βMOBi,t5
,NT
t5,I,r, β
NT,MOAi,t5t5,I,r
)
MOAi,t5 if β ∈ [βNT,MOAi,t5t5,I,r
, 2]
(18)
where
βMOBi,t5
,NT
t5,I,r=
Bi,t5 −∆
v(19)
βNT,MOAi,t5t5,I,r
=Ai,t5 −∆
v
are the critical thresholds that solve ϕt5,r(MOBi,t5) = ϕt5,r(NT ) and ϕt5,r(NT ) = ϕt5,r(MOAi,t5),
respectively. The optimal trading strategies and β thresholds for an uninformed traders are similar
but the conditioning set does not include the signal on v:
xt5,U,r(β|Lt4) =
MOBi,t5 if β ∈ [0, β
MOBi,t5,NT
t5,U,r)
NT if β ∈ [βMOBi,t5
,NT
t5,U,r, β
NT,MOAi,t5t5,U,r
)
MOAi,t5 if β ∈ [βNT,MOAi,t5t5,U,r
, 2]
(20)
where
βMOBi,t5
,NT
t5,U,r=
Bi,t5 − Er−1[∆|Lt4 ]
v(21)
βNT,MOAi,t5t5,U,r
=Ai,t5 − Er−1[∆|Lt4 ]
v
Once we know the β ranges associated with each strategy, we compute the submission prob-
abilities associated with each optimal order at t5 using the distribution of β. At time t4 these
probabilities are the execution probabilities for limit orders at the best bid and ask, Bi,t4 and Ai,t4
13For instance, an informed trader would post a MOA1 only if the payoff is positive and thus outperforms the NTpayoff of zero, i.e, βv + ∆−A1 > 0 or β > A1−∆
v.
51
respectively at time t5:
Prr(θLOBi,t4 |Lt3 , v) =
∫β∈[0, β
MOBi,t4,NT
t5,I,r
) n(β) dβ where i indexes the best bid and if qBi,t4t3
= 0
0 otherwise
(22)
Prr(θLOAi,t4 |Lt3 , v) =
∫β∈[βNT,MOAi,t4t5,I,r
, 2] n(β) dβ where i indexes the best ask and if q
Ai,t4t3
= 0
0 otherwise
(23)
where qBi,t4t3
= 0 and qAi,t4t3
= 0 means that the incoming limit order book from time t3 is empty at
the best bid and ask at time t4. The execution probabilities of uninformed at the best bid and the
best ask:
Prr(θLOBi,t4 |Lt3) =
∫β∈[0, β
MOBi,t4,NT
t5,U,r
) n(β) dβ where i indexes the best bid and if qBi,t4t3
= 0
0 otherwise
(24)
Prr(θLOAi,t4 |Lt3) =
∫β∈[βNT,MOAi,t4t5,U,r , 2
] n(β) dβ where i indexes the best ask and if qAi,t4t3
= 0
0 otherwise
(25)
where n(·) is the truncated normal density function. At t4 there is only one period before the end
of the trading game. Thus, the execution probability of a limit order is positive if and only if the
order is posted at the best price on its own side of the market (Pi,tj ), and if there are no limit
orders already standing in the limit order book at that price at the time the limit order is posted
(qBi,t4t3
= 0 and qAi,t4t3
= 0).
Having obtained the execution probabilities for limit orders at t4, we next derive the optimal
order-submission strategies at t4. The book can open in many different ways at t4 depending on the
prior path of the trading game. As the payoffs of both limit and market orders are functions of β,
we rank all the payoffs of adjacent optimal strategies in terms of β and equate them to determine
the β thresholds at time t4.14
Consider for example, a path of the game such that the book opens empty; so both limit and
14Recall that the upper envelope only includes strategies that are optimal.
52
market orders are optimal strategies at t4. For an informed trader, these strategies are:
xt4,I,r(β|Lt3 , v) =
MOB2,t4 if β ∈ [0, βMOB2,t4 ,LOA1,t4t4,I,r
)
LOA1,t4 if β ∈ [βMOB2,t4 ,LOA1,t4t4,I,r
, βLOA1,t4 ,LOA2,t4t4,I,r
)
LOA2,t4 if β ∈ [βLOA1,t4 ,LOA2,t4t4,I,r
, βLOA2,t4 ,NT
t4,I,r)
NT if β ∈ [βLOA2,t4 ,NT
t4,I,r, β
NT,LOB2,t4t4,I,r
)
LOB2,t4 if β ∈ [βNT,LOB2,t4t4,I,r
, βLOB2,t4 ,LOB1,t4,t4t4,I,r
)
LOB1,t4 if β ∈ [βLOB2,t4 ,LOB1,t4t4,I,r
, βLOB1,t4 ,MOA2,t4t4,I,r
)
MOA2,t4 if β ∈ [βLOB1,t4 ,MOA2,t4t4,I,r
, 2]
(26)
and for an uninformed trader the optimal strategies are qualitatively similar but with different
values for the β thresholds given the uninformed investor’s different information.15 As the payoffs
of both limit and market orders are functions of β, we can rank all the payoffs of adjacent optimal
strategies in terms of β and equate them to determine the β thresholds at t4. For example, for the
first threshold we have:
βMOB2,t4 ,LOA1,t4t4,I,r
= β ∈ R s.t. ϕIt4,r (MOB2,t4 | v, β,Lt3) = ϕIt4,r (LOA1,t4 | v, β,Lt3) (27)
and we obtain the other thresholds similarly.
The next step is to use the β thresholds together with the truncated Normal cumulative dis-
tribution N(�) for β to derive the probabilities of the optimal order-submission strategies at each
possible node of the extensive form of the game at t4. For example, the submission probability of
LOAi,t4 is:
Prr[LOA1,t4 |Lt3 , v] = N(βLOA1,t4 ,LOA2,t4t4,I,r
|Lt3 , v)− N(βMOB2,t4 ,LOA1,t4t4,I,r
|Lt3 , v) (28)
and the submission probabilities of the equilibrium strategies can be obtained in a similar way.
Next, given the market-order submission probabilities at t4 (which are the execution probabilities
15If the book opened with some liquidity on any level of the book, the equilibrium strategies would be different.For example, if the book opened with a LOA1 then no limit orders on the ask side would be equilibrium strategies.
53
of limit orders at t3), we can solve the optimal orders at t3 and recursively we can then solve the
model by backward induction back to time t1.
At each node of the trading game, the algorithm considers all feasible orders that traders may
choose. Off-equilibrium orders are those that are never chosen as part of the optimal trading
strategies. Suppose that in round r an order that is off-equilibrium in round r− 1 is considered for
time tj . For example, consider in round r the path of the trading game ending with LOA1,t3 formed
by the sequence of orders: {MOA2,t1 ,MOB2,t2 , LOA1,t3}, where LOA1,t3 was not an equilibrium
strategy at t3 in round r− 1 and where MOA2,t1 and MOB2,t2 are equilibrium strategies at times
t1 and t2 respectively. Within the convergence process, for each strategy which is reconsidered
in the subsequent round, uninformed traders generally use their previous round beliefs. For an
off-equilibrium strategy at tj in r − 1, however, they cannot use their r − 1 updated belief for
that time and therefore they use their most recent equilibrium belief up to tj still for round r − 1.
Considering the example above, uninformed traders cannot use their updated belief conditional
on the sequence of orders {MOA2,t1 ,MOB2,t2 , LOA1,t3} at t3 for round r − 1 as LOA1,t3 was
not an equilibrium strategy. Therefore we assume that for this off-equilibrium belief, uninformed
traders use the most updated equilibrium belief before t3, formed by using the sequence of orders
{MOA2,t1 ,MOB2,t2}. If instead in round r − 1, MOB2,t2 is still not an equilibrium strategy at
t2, we assume that uninformed traders use their belief at t1 conditional on MOA2,t1 . Finally, if
neither MOA2,t1 were an equilibrium strategy at t1 we assume that traders use their unconditional
prior belief.
We allow for both pure and mixed strategies in our Perfect Bayesian Nash equilibrium. When
different orders have equal expected payoffs, we assume that traders randomize with equal probabil-
ities across all such optimal orders. By construction, the expected payoffs of two different strategies
are the same in correspondence of the β thresholds; however because we are considering single points
in the support of the β distribution, the probability associated with any strategy that corresponds
to those specific points is equal to zero. This means that mixed strategies that emerge in cor-
respondence of the β thresholds, although feasible, have zero probability. Mixed strategies may
also emerge in the framework in which informed traders have a fixed neutral private-value factor
54
β = 1 (section 2.1). More specifically it may happen that the payoffs of two perfectly symmetrical
strategies of Iv0 are the same, and in this case Iv0 randomizes between these two strategies.
RE beliefs for a Perfect Bayesian Nash equilibrium are obtained by solving the model recursively
for multiple rounds. In particular, the asset-value probabilities from round r = 1 above are used as
the priors to solve the model in round r = 2 (i.e., the round 1 probabilities are used in place of the
unconditional priors used in round 1).16 The asset-value probabilities from round r = 2 are then
used as the priors in round r = 3 and so on. We continue the iteration until the updating process
converges to a fixed point, which are the REE beliefs. In particular, the recursive process has
converged to the RE beliefs when uninformed traders do not revise their asset-value beliefs. Opera-
tionally, we consider convergence to the RE beliefs to have occurred when the execution-contingent
conditional probabilities πv, rtj, πv0, r
tjand π
v, rtj
in round r are almost equal to the corresponding
probabilities from round r − 1:
πv, ∗tj when∣∣∣πv, rtj
− πv, r−1tj
∣∣∣ < 10−7
πv0, ∗tj
when∣∣∣πv0, rtj− πv0, r−1
tj
∣∣∣ < 10−7
πv, ∗tj
when∣∣∣πv, rtj
− πv, r−1tj
∣∣∣ < 10−7
(29)
The fixed point is such that conditional on the most recent pieces of information, uninformed
traders can extract from the limit order book, they do not wish to revise their beliefs on πvtj , πv0tj
and πvtj
. A fixed-point solution to this recursive algorithm is an equilibrium in our model.
16In the second round of solutions we again solve the full 5-period model.
55
6 Appendix B: Additional numerical results
The tables is this section provide additional information on the execution probabilities of limit or-
ders for informed investor with positive, neutral and negative signals, (Iv,Iv0 ,Iv¯) and for uninformed
traders. The tables also report the asset value expectations of the uninformed investor at time t2
after observing all the possible buy orders submissions at time t1 (the expectations for sell orders
are symmetric with respect to 1). Table B1 reports results for the model specification in which
only uninformed traders have a random private value factor, Table B2 instead reports results for
the model in which both the informed and the uniformed traders have private-value motives.
Table B1: Order Execution Probabilities and Asset-Value Expectation for Informed Traders withβ = 1 and Uninformed Traders with β ∼ Tr[N (µ, σ2)]. This table reports results for two different values ofthe informed-investor arrival probability α (0.8 and 0.2) and for two different values of the asset-value volatility δ (0.16and 0.02). σ = 1.5. For each set of parameters, the first four columns report the equilibrium limit order probabilitiesof executions for informed traders with positive, neutral and negative signals, (Iv,Iv0 ,Iv
¯) and for uninformed traders
(U). The fifth column (Uncond.) reports the unconditional order-execution probabilities in the market. Next, thecolumns report conditional and unconditional future order execution probabilities and the asset-value expectationsof an uniformed investor at time t2 after observing different order submissions at time t1.
δ = 0.16 δ = 0.02
Iv Iv0 Iv¯
U Uncond. Iv Iv0 Iv¯
U Uncond.
PEX(LOA2|·) 0.955 0.175 0.055 0.395 0.395 0.180 0.229 0.170 0.193 0.193PEX(LOA1|·) 0.989 0.125 0.078 0.397 0.397 0.323 0.323 0.323 0.323 0.323PEX(LOB1|·) 0.078 0.125 0.989 0.397 0.397 0.323 0.323 0.323 0.323 0.323PEX(LOB2|·) 0.055 0.175 0.955 0.395 0.395 0.170 0.229 0.180 0.193 0.193
α = 0.8E[v|LOB1 |·] 1.160 1.000E[v|LOB2 |·] 1.083 1.013E[v|MOA1 |·]E[v|MOA2 |·] 1.000 1.000
PEX(LOA2|·) 0.651 0.487 0.394 0.511 0.511 0.514 0.499 0.476 0.496 0.496PEX(LOA1|·) 0.886 0.766 0.717 0.789 0.789 0.792 0.792 0.790 0.791 0.791PEX(LOB1|·) 0.717 0.766 0.886 0.789 0.789 0.790 0.792 0.792 0.791 0.791PEX(LOB2|·) 0.394 0.487 0.651 0.511 0.511 0.476 0.499 0.514 0.496 0.496
α = 0.2E[v|LOB1 | ·] 1.026 1.000E[v|LOB2 | ·] 1.013 1.009E[v|MOA1 | ·]E[v|MOA2 | ·] 1.000 1.000
56
Table B2: Order Execution Probabilities and Asset-Value Expectation for Informed and UninformedTraders both with β ∼ Tr[N (µ, σ2)]. This table reports results for two different values of the informed-investorarrival probability α (0.8 and 0.2) and for two different values of the asset-value volatility δ (0.16 and 0.02). σ = 1.5.For each set of parameters, the first four columns report the equilibrium limit order probabilities of executions forinformed traders with positive, neutral and negative signals, (Iv,Iv0 ,Iv
¯) and for uninformed traders (U). The fifth
column (Uncond.) reports the unconditional order-execution probabilities in the market. Next, the columns reportconditional and unconditional future order execution probabilities and the asset-value expectations of an uniformedinvestor at time t2 after observing different order submissions at time t1.
δ = 0.16 δ = 0.02
Iv Iv0 Iv¯
U Uncond. Iv Iv0 Iv¯
U Uncond.
PEX(LOA2|·) 0.644 0.502 0.410 0.519 0.135 0.502 0.487 0.472 0.487 0.116PEX(LOA1|·) 0.913 0.834 0.702 0.817 0.392 0.849 0.837 0.824 0.836 0.470PEX(LOB1|·) 0.702 0.834 0.913 0.817 0.392 0.824 0.837 0.849 0.836 0.470PEX(LOB2|·) 0.410 0.502 0.644 0.519 0.135 0.472 0.487 0.502 0.487 0.116
α = 0.8E[v|LOB1 |·] 0.996 1.000E[v|LOB2 |·] 0.944 0.999E[v|MOA1 |·]E[v|MOA2 |·] 1.156
PEX(LOA2|·) 0.525 0.494 0.470 0.496 0.402 0.490 0.487 0.483 0.487 0.394PEX(LOA1|·) 0.853 0.833 0.813 0.833 0.737 0.839 0.837 0.834 0.837 0.745PEX(LOB1|·) 0.813 0.833 0.853 0.833 0.737 0.834 0.837 0.839 0.837 0.745PEX(LOB2|·) 0.470 0.494 0.525 0.496 0.402 0.483 0.487 0.490 0.487 0.394
α = 0.2E[v|LOB1 |·] 1.003 1.000E[v|LOB2 |·] 0.996 1.000E[v|MOA1 |·]E[v|MOA2 |·] 1.160
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